solve the problem using system of equations

Online Systems of Equations Solver

Solve equations and systems of equations with wolfram|alpha, a powerful tool for finding solutions to systems of equations and constraints.

Wolfram|Alpha is capable of solving a wide variety of systems of equations. It can solve systems of linear equations or systems involving nonlinear equations, and it can search specifically for integer solutions or solutions over another domain. Additionally, it can solve systems involving inequalities and more general constraints.

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Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask about solving systems of equations.

• solve y = 2x, y = x + 10
• solve system of equations {y = 2x, y = x + 10, 2x = 5y}
• y = x^2 - 2, y = 2 - x^2
• solve 4x - 3y + z = -10, 2x + y + 3z = 0, -x + 2y - 5z = 17
• solve system {x + 2y - z = 4, 2x + y + z = -2, z + 2y + z = 2}
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What are systems of equations?

A system of equations is a set of one or more equations involving a number of variables..

The solutions to systems of equations are the variable mappings such that all component equations are satisfied—in other words, the locations at which all of these equations intersect. To solve a system is to find all such common solutions or points of intersection.

Systems of linear equations are a common and applicable subset of systems of equations. In the case of two variables, these systems can be thought of as lines drawn in two-dimensional space. If all lines converge to a common point, the system is said to be consistent and has a solution at this point of intersection. The system is said to be inconsistent otherwise, having no solutions. Systems of linear equations involving more than two variables work similarly, having either one solution, no solutions or infinite solutions (the latter in the case that all component equations are equivalent).

More general systems involving nonlinear functions are possible as well. These possess more complicated solution sets involving one, zero, infinite or any number of solutions, but work similarly to linear systems in that their solutions are the points satisfying all equations involved. Going further, more general systems of constraints are possible, such as ones that involve inequalities or have requirements that certain variables be integers.

Solving systems of equations is a very general and important idea, and one that is fundamental in many areas of mathematics, engineering and science.

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Unit 5: Systems of linear equations and inequalities

About this unit.

If one line is useful, let's see what we can do with two lines. In this unit, we learn how to write systems of equations, solve those systems, and interpret what those solutions mean in a real-world contexts.

Introduction to systems of equations

• Systems of equations: trolls, tolls (1 of 2) (Opens a modal)
• Systems of equations: trolls, tolls (2 of 2) (Opens a modal)
• Testing a solution to a system of equations (Opens a modal)
• Systems of equations with graphing: y=7/5x-5 & y=3/5x-1 (Opens a modal)
• Systems of equations with graphing: exact & approximate solutions (Opens a modal)
• Solutions of systems of equations Get 3 of 4 questions to level up!
• Systems of equations with graphing Get 3 of 4 questions to level up!

Number of solutions to systems of equations

• Number of solutions to a system of equations (Opens a modal)
• Number of solutions to a system of equations graphically (Opens a modal)
• Number of solutions to a system of equations graphically Get 3 of 4 questions to level up!

Write a system of equations from a table or graph

• Intercepts from a table (Opens a modal)
• Calculating slope from tables (Opens a modal)
• Intercepts of lines review (x-intercepts and y-intercepts) (Opens a modal)
• Worked example: slope from graph (Opens a modal)
• Write systems of two linear equations given a table Get 3 of 4 questions to level up!
• Write systems of two linear equations given a graph Get 3 of 4 questions to level up!

Systems of equations from context

• Setting up a system of equations from context example (pet weights) (Opens a modal)
• Setting up a system of linear equations example (weight and price) (Opens a modal)
• Creating systems in context Get 3 of 4 questions to level up!
• Estimate graphically with systems of equations in real-world problems Get 3 of 4 questions to level up!

Interpreting points relative to a system of equations

• Interpreting points in context of graphs of systems (Opens a modal)
• Interpret points relative to a system Get 3 of 4 questions to level up!

Solving systems of equations with substitution

• Systems of equations with substitution: potato chips (Opens a modal)
• Systems of equations with substitution: -3x-4y=-2 & y=2x-5 (Opens a modal)
• Substitution method review (systems of equations) (Opens a modal)
• Systems of equations with substitution Get 3 of 4 questions to level up!

Solving systems of equations with elimination

• Systems of equations with elimination: King's cupcakes (Opens a modal)
• Elimination strategies (Opens a modal)
• Systems of equations with elimination: x-4y=-18 & -x+3y=11 (Opens a modal)
• Systems of equations with elimination: potato chips (Opens a modal)
• Systems of equations with elimination (and manipulation) (Opens a modal)
• Elimination method review (systems of linear equations) (Opens a modal)
• Combining equations Get 3 of 4 questions to level up!
• Elimination strategies Get 3 of 4 questions to level up!
• Systems of equations with elimination Get 3 of 4 questions to level up!
• Systems of equations with elimination challenge Get 3 of 4 questions to level up!

Equivalent systems of equations

• Why can we subtract one equation from the other in a system of equations? (Opens a modal)
• Worked example: equivalent systems of equations (Opens a modal)
• Worked example: non-equivalent systems of equations (Opens a modal)
• Reasoning with systems of equations (Opens a modal)
• Equivalent systems of equations review (Opens a modal)
• Reasoning with systems of equations Get 3 of 4 questions to level up!

Systems of equations word problems

• Age word problem: Imran (Opens a modal)
• Age word problem: Ben & William (Opens a modal)
• Age word problem: Arman & Diya (Opens a modal)
• System of equations word problem: walk & ride (Opens a modal)
• System of equations word problem: no solution (Opens a modal)
• System of equations word problem: infinite solutions (Opens a modal)
• Systems of equations with elimination: TV & DVD (Opens a modal)
• Systems of equations with elimination: apples and oranges (Opens a modal)
• Systems of equations with substitution: coins (Opens a modal)
• Systems of equations with elimination: coffee and croissants (Opens a modal)
• Systems of equations: FAQ (Opens a modal)
• Age word problems Get 3 of 4 questions to level up!
• Systems of equations word problems Get 3 of 4 questions to level up!
• Systems of equations word problems (with zero and infinite solutions) Get 3 of 4 questions to level up!

Graphing systems of two-variable inequalities

• Intro to graphing systems of inequalities (Opens a modal)
• Graphing systems of inequalities (Opens a modal)
• Systems of inequalities graphs Get 3 of 4 questions to level up!

System of Equations Calculator

What is a system of linear equations, how to solve a system of equations, example: using the system of equations solver, example: solving systems of equations by gaussian elimination.

Welcome to the system of equations calculator , where we will learn how to solve a system of linear equations . Our handy calculator will quickly find the solution to any problem you give it, and, if there are an infinite number of solutions, it will even tell you what they look like ! The system of equations solver uses the so-called Gaussian elimination method , but this is not the only method, so below we present five different answers to the question "How to solve a system of equations?"

Let's not waste a second longer and get to it, shall we?

Remember all those riddles on Facebook or Instagram , you know, the ones where three apples are equal to 30, an apple and two bananas are equal to 18, and a banana minus a coconut equal to two, and you had to calculate how much the apple, banana, and coconut are worth? That is what the mathematicians call a system of linear equations . " But how? Mathematicians don't use apples and bananas, do they? " Well, they too like to keep the doctor away and bite into an apple from time to time, but you're right, they don't calculate in apples . However, it doesn't make any difference if you right " Three apples equal to 30 ," or 3x = 30 .

The x that appeared above is what we call a variable . It denotes a number or element that we don't know the value of, but that we do know something about. In our case, we know that three apples equal to 30 , but the apple is simply a variable, like x , as we don't know the value of it. In essence, " what is the solution to the system of equations... " is the same as " give me the value of an apple (or x ) that satisfies ..." To be honest, we know that most scientists would love to use bananas instead of x 's, but they're just insecure about their drawing skills .

" But what the heck does linear mean? " We say that an equation is linear if its variables (be they x 's or coconuts) are to the first power. This means that, for instance, they are not squared x² as in quadratic equations, or the denominator of a fraction, or under a square root. They can, however, be multiplied by any number, just as we had the 3 in our 3x = 30 equation. This applies to all the variables in an equation . For example, the equation -2x + 14y - 0.3z = 0 is linear, but 10x - 7y + z² = 1 isn't.

💡 Check our quadratic formula calculator to understand what a quadratic equation is!

Lastly, if we have a few equations to solve together, then we call them a system of equations . We denote it by drawing a curly bracket (or a rotated set of mustache, whichever you prefer) to the left of them. This means that we're only interested in solutions to all the equations in the system . If we find values that work for the first equation but don't for the second, then we don't call that a solution.

There are many different ways to solve a system of linear equations. Let's briefly describe a few of the most common methods.

1. Substitution

The first method that students are taught, and the most universal method , works by choosing one of the equations, picking one of the variables in it, and making that variable the subject of that equation . Then, we use this rearranged equation and substitute it for every time that variable appears in the other equations. This way, those other equations now have one variable less , which makes them easier to solve.

For example, if we have an equation 2x + 3y = 6 and want to get x from it, then we start by getting rid of everything that doesn't contain x from the left-hand side . To do this, we have to subtract 3y from both sides (because we have that expression on the left). This means that the left side will be 2x + 3y - 3y , which is simply 2x , and the right side will be 6 - 3y . In other words, we have transformed our equation into 2x = 6 - 3y .

Since we want to get x , and not 2x , we still need to get rid of the 2 . To do this, we divide both sides by 2. This way, on the left, we get (2x) / 2 , which is just x , and, on the right, we have (6 - 3y) / 2 , which is 3 - 1.5y . All in all, we obtained x = 3 - 1.5y , and we can use this new formula to substitute 3 - 1.5y in for every x in the other equations.

2. Elimination of variables

Solving systems of equations by elimination means that we're trying to reduce the number of variables in some of the equations to make them easier to solve . To do this, we start by transforming two equations so that they look similar. To be precise, we want to make the coefficient (the number next to a variable) of one of the equations variables the opposite of the coefficient of the same variable in another equation . We then add the two equations to obtain a new one, which doesn't have that variable, and so it is easier to calculate.

For example, if we have a system of equations,

2x + 3y = 6 , and

4x - y = 3 ,

then we can try to make the coefficient of x in the first equation to be the opposite of the coefficient in the second equation. In our case, this means that we want to transform the 2 into the opposite of 4 , which is -4 . To do this, we need to multiply both sides of the first equation by -2 , since 2 × (-2) = -4 . This changes the first equation into

2x × (-2) + 3y × (-2) = 6 × (-2) ,

-4x - 6y = -12 .

Now we can add this equation to the second one (the 4x - y = 3 ) by adding the left side to the left side and the right to the right. This gives

4x - y + (-4x - 6y) = 3 + (-12) ,

We've obtained a new equation with just one variable, which means that we can easily solve y . We can then substitute that number into either of the original equations to get x .

3. Gaussian elimination method

This is the method used by our system of equations calculator. Named after a German mathematician Johann Gauss, it is an algorithmic extension of the elimination method presented above. In the case of just two equations, it is exactly the same thing. However, solving systems of equations by regular elimination gets trickier and trickier with more and more equations and variables. That's where the Gaussian elimination method comes in.

Let's say that we have four equations with four variables . To find the solution to our system, we want to try to get the values of our variables one by one by eliminating all the other consecutively. To do this, we take the first equation and the first of the variables . We use its coefficient to eliminate all the occurrences of that particular variable in the other three equations , just as we did in the regular elimination. This way, we are left with the first equation the same as it was and three equations, now each with only three variables .

We now look at the first equation, give it a thumbs-up, and leave it as it is until the very end . We repeat the process for the other three equations. In other words, we take the second variable and its coefficient from the second equation to eliminate all occurrences of that variable in the last two equations. This leaves us with the first equation having four variables, the second having three, and the last two having only two variables .

Next, we declare the second equation to be nice and pretty and leave it be. We move on to the two remaining equations and take the third variable and its coefficient in the third equation to eliminate that variable from the fourth equality.

In the end, we obtain a system of four equations, in which the first has four variables, the second has three, the third has two, and the last has only one . This means that we can easily get the value of the fourth variable from the fourth equation (since it has no other variables). We then substitute that value to the third equation and get the value of the third variable (since it now has no other variables), and so on.

4. Graphical representation

Arguably the least used method, but a method nonetheless. It takes each of the equations in our system and translates them to a function . The points on the graph of such a function correspond to the coordinates that satisfy that equation. Therefore, if we want to solve a system of linear equations, then it is enough to find all the points where the line cross on the graph , i.e., the coordinates that satisfy all of the equations.

It can be, however, tricky. If we have just two equations and two variables, then the functions are lines on a two-dimensional plane. Therefore, we just need to find the point where those two lines cross .

For three variables, the functions are now in a three-dimensional space, and are no longer lines but planes . This means that we would have to draw three planes (which is tricky in itself) and then also find where those planes cross. And, if you think that's difficult, try to imagine four variables and four dimensions . If it comes to you naturally, please contact us, and we'll direct you to the nearest Nobel prize-type project or a neurologist for a thorough head check.

🙋 By describing them using the slope-intercept form , you can easily find the intersection between two lines. Read more about it in our slope intercept form calculator .

5. Cramer's rule

A fairly easy and very straightforward way to solve a system of linear equations. It does, however, require a good understanding of matrices and their determinants . As an encouragement, let us mention that it doesn't need any substitution, no playing around with the equations, it's just the good old basic arithmetics . For example, for a system of three equations with three variables, we plug in the coefficients from those equations to form four three-by-three matrices and calculate their determinants ( what is a determinant? ). We finish by dividing the appropriate values that we've just obtained to get the final solution.

Let's look at one of those picture riddles , and try to solve it with our system of equations calculator .

The first thing we have to do is write all the tasty sweets as letter variables. We know the expression we'll get will be far from an eye candy , but mathematicians don't have much taste . Okay, let's get to work and leave the puns for dessert .

Our riddle has three symbols – the doughnut, the cookie, and the candy. We don't know the value of any of them, so we'll need three variables – one for each of the pictures. It is customary to use letters like x , y , and z , but feel free to use others if you please. We will denote the doughnut by x , the cookie by y , and the candy by z . This allows us to write the above riddle in the form:

x + x + x = y

y + y - z = 25

z + z - x = 16 .

So, what is the solution to the system of equations? Now, hold your horses. First of all, we'll try to simplify each of the three expressions before we even think about how to solve this system of equations. Note that our system of equations solver doesn't use the formulas in the form that we have now . In particular, it doesn't have any variables on the right side of the = sign, as we have in the first expression. So, we do indeed have to do some work first.

We take each of the equations and move all the variables to the left-hand side . Then, we add together all summands with the same variable ( x , y , or z ) in that equation. Finally, we write the summands that we got in alphabetical order , in terms of the variables. This means that we write the expression with x first, then the expression with y , and then the one with z .

In our case, this means that we have to first move the y in the first equation from right to left. To do this, we subtract y from both sides of the equality. This gives

x + x + x - y = y - y ,

x + x + x - y = 0 .

The whole system now looks like this:

x + x + x - y = 0

z + z - x = 16

Now, we add up all the summands that contain the same variable . This means that in the first equation, we add the three x 's, in the second, we add the two y 's, and in the third, we add the two z 's. We obtain

2y - z = 25

2z - x = 16 .

Remember, that when we write 3x , we mean 3 × x , or "three copies of x" . Now, we write the variables in alphabetical order . The first two equations already have the form we want, but in the last, we need to move the expression with x before the one with z . This gives

-x + 2z = 16

Observe that, at first glance, this doesn't look like the expression we have in the system of equations calculator . It is, however, just that. For example, the first equation doesn't have any z 's in it. But recall that "no z 's" means "zero copies of z ." We can, therefore, write the missing variables with coefficients 0. In this way, we obtain

3x - y + 0z = 0

0x + 2y - z = 25

-x + 0y + 2z = 16

Now, this is more like it – this is just the form of the system of equations solver! Just to be sure, remember that when we have no number in front of a variable, then it is a customary way of saying that the number is 1. For example, the -y in the first equation is, in fact -1y .

Finally, we need to identify what data we need to take from the system we've obtained and where to put it in the system of equations calculator . Well, let's look at the first equality we have and the top one from the solver and compare them:

a₁x + b₁y + c₁z = d₁

The correspondence is just as it looks: a₁ is the number next to x in the equation, b₁ is the one next to y , c₁ is the one next to z , and d₁ is the number we have on the right. In our case, this means that we should put a₁ = 3 , b₁ = -1 , c₁ = 0 , and d₁ = 0 . We repeat this with the second and the third equation: a₂ = 0 , b₂ = 2 , c₂ = -1 , d₂ = 25 , a₃ = -1 , b₃ = 0 , c₃ = 2 , d₃ = 16 . Once we give all those numbers, the system of equations solver will give us the solution . In the next section , we describe how it does that, step by step .

Dealing with cookies and doughnuts is all fun and games, but let's now try to burn some of those sugary calories by describing how to solve the system of equations we've obtained in the above section :

We want to leave the first equation as it is , since it has a non-zero coefficient next to the variable x . We will, however, use that coefficient to get rid of x 's in the other equations . Observe that we don't have to worry about the second one, because its x coefficient is zero. To deal with the third, we will eliminate the -x from it by first transforming it into the opposite of 3x from the first equation. In fact, it is enough to multiply both sides of the third equation by 3 .

-3x + 0y + 6z = 48

Now we have opposite numbers next to x in the first and the last equality, we add the two expressions together

(3x - y + 0z) + (-3x + 0y +6z) = 0 + 48 ,

0x -y + 6z = 48 .

We can now replace the third equation with the one we've just obtained , to get

0x - y + 6z = 48

What we've gained by this is that the two last expressions have no x in them, and it is always easier to solve a system of linear equations with two variables instead of three.

The next step in the Gaussian elimination method is to repeat the same process for the last two equations . In essence, we will use the non-zero coefficient of the y in the second equality to get rid of the y from the last one. As we've done above, we start by transforming the -y into the opposite of 2y , i.e., into -2y . To do this, it is enough to multiply both sides of the last equation by 2.

0x - 2y + 12z = 96

We can now add the two last equations to get

(0x + 2y - z) + (0x - 2y + 12z) = 25 + 96 ,

0x + 0y + 11z = 121 .

Time to replace the third equation

This is the end-form of the system of equations that we get from the Gaussian elimination method . Now it is so much easier to solve the system of linear equations. How so? Well, let's begin with the last equality. It has only one variable with a non-zero coefficient, namely z . We can forget about the zero terms, which gives us

11z = 121 ,

and that means that we must have z = 11 . Now that we know what the first part of the solution to the system of equations is, we can use this knowledge to substitute that number for z in the other two equations :

3x - y + 0 = 0

0x + 2y - 11 = 25 ,

0x + 2y = 36 .

Now we have the second equation with only one variable with a non-zero coefficient. If we forget about the zero terms, we'll get

and therefore, we must have y = 18 . Again, we substitute that number for y in the first equation :

3x - 18 = 0 ,

which gives

and that means that x = 6 . All in all, we've managed to solve the system of linear equations and find the solution to be

If we now look at our picture riddle, all this solving the system of equation by elimination leads us to an answer that a doughnut equals 6 , a cookie equals 18 , and a candy equals 11 .

A piece of cake, wasn't it?

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System of equations calculator

Enter coefficients of your system into the input fields. Leave cells empty for variables, which do not participate in your equations. To input fractions use / : 1/3 .

• Show how to input the following system: 2x-2y+z=-3 x+3y-2z=1 3x-y-z=2

This calculator solves Systems of Linear Equations with steps shown, using Gaussian Elimination Method , Inverse Matrix Method , or Cramer's rule . Also you can compute a number of solutions in a system (analyse the compatibility) using Rouché–Capelli theorem .

• Leave extra cells empty to enter non-square matrices.
• decimal (finite and periodic) fractions: 1/3 , 3.14 , -1.3(56) , or 1.2e-4
• mathematical expressions : 2/3+3*(10-4) , (1+x)/y^2 , 2^0.5 (= 2 ) , 2^(1/3) , 2^n , sin(phi) , cos(3.142rad) , a_1 , or (root of x^5-x-1 near 1.2)
• matrix literals: {{1,3},{4,5}}
• operators: + , - , * , / , \ , ! , ^ , ^{*} , , , ; , ≠ , = , ⩾ , ⩽ , > , and
• functions: sqrt , cbrt , exp , log , abs , conjugate , arg , min , max , gcd , rank , adjugate , inverse , determinant , transpose , pseudoinverse , trace , cos , sin , tan , cot , cosh , sinh , tanh , coth , arccos , arcsin , arctan , arccot , arcosh , arsinh , artanh , arcoth , derivative , factor , and resultant
• units: rad , deg
• pi , e , i — mathematical constants
• k , n — integers
• I or E — identity matrix
• X , Y — matrix symbols
• Use ↵ Enter , Space , ← ↑ ↓ → , Backspace , and Delete to navigate between cells, Ctrl ⌘ Cmd + C / Ctrl ⌘ Cmd + V to copy/paste matrices.
• Drag-and-drop matrices from the results, or even from/to a text editor.
• To learn more about matrices use Wikipedia .

System of Equations

In mathematics, a system of equations, also known as a set of simultaneous equations or an equation system, is a finite set of equations for which we sought common solutions. In systems of equations, variables are related in a specific way in each equation. i.e., the equations can be solved simultaneously to find a set of values of variables that satisfies each equation.

System of linear equations finds applications in our day-to-day lives in modelling problems where the unknown values can be represented in form of variables. Solving system of equations involves different methods such as substitution, elimination, graphing, etc. Let us look into each method in detail.

What is a System of Equations?

In algebra, a system of equations comprises two or more equations and seeks common solutions to the equations. "A system of linear equations is a set of equations which are satisfied by the same set of values of variables."

System of Equations Example

A system of equations as discussed above is a set of equations that seek a common solution for the variables included. The following set of linear equations is an example of the system of equations:

• 2x - y = 12
• x - 2y = 48

Note that the values x = -8 and y = -28 satisfy each of these equations and hence the pair (x, y) = (-8, -28) is the solution of the above system of equations. But how to solve system of equations? Let us see.

Solutions of System of Equations

Solution of system of equations is the set of values of variables that satisfies each linear equation in the system. The main reason behind solving an equation system is to find the value of the variable that satisfies the condition of all the given equations true. There systems of equations are classified into 3 types depending on their number of solutions:

• Linear System with "Unique solution"
• Linear System with "No solution"
• Linear System with "Infinitely many solutions"

We know that every linear equation represents a line on the coordinate plane. In this perception, the above figure should give more sense to understanding the different types of solutions of system of equations.

Unique Solution of a System of Equations

A system of equations has unique solution when there is only set of variables exist that satisfy each equation in the system. In terms of graphs, a system with a unique solution has lines (representing the equations) that intersect (at one point).

No Solution

A system of equations has no solution when there exists no set of variables that satisfy each linear equation in the system. If we graph that kind of system, the resulting lines will be parallel to each other.

Infinite Many Solutions

A system of equations can have infinitely many solutions when there exist infinite sets of variables that satisfy each equation. In such cases, the lines corresponding to the linear equations would overlap each other on the graph. i.e., both equations represent the same line. Since a line has infinite points on it, each point on it becomes a solution of the system.

Solving System of Equations

Solving a system of equations means finding the values of the variables used in the set of equations. Any system of equations can be solved in different methods.

• Substitution Method
• Elimination Method
• Graphical Method
• Cross-multiplication method

To solve a system of equations in 2 variables, we need at least 2 equations. Similarly, for solving a system of equations in 3 variables, we will require at least 3 equations. Let us understand 3 ways to solve a system of equations given the equations are linear equations in two variables.

☛ Also Check: Solving System of Linear Equations

Solving System of Linear Equations By Substitution Method

For solving the system of equations using the substitution method given two linear equations in x and y, in one of the equations, express y in terms x in one of the equations and then substitute it in the other equation.

Example: Solve the system of equations using the substitution method.

3x − y = 23 → (1)

4x + 3y = 48 → (2)

From (1), we get:

y = 3x − 23 → 3

Plug in y in (2),

4x + 3 (3x − 23) = 48

13x − 69 = 48

Now, plug in x = 9 in (1)

y = 3 × 9 − 23 = 4

Hence, x = 9 and y = 4 is the solution of the given system of equations.

Solving System of Equations By Elimination Method

Using the elimination method to solve the system of equations, we eliminate one of the unknowns, by multiplying equations by suitable numbers, so that the coefficients of one of the variables become the same.

Example: Solve the following system of linear equations by elimination method.

2x + 3y = 4 → (1) and 3x + 2y = 11 → (2)

The coefficients of y are 3 and 2; LCM (3, 2) = 6

Multiplying Equation (1) by 2 and Equation (2) by 3, we get

4x + 6y = 8 → (3)

9x + 6y = 33 → (4)

On subtracting (3) from (4), we get

Plugging in x = 5 in (2) we get

15 + 2y = 11

Hence, x = 5, y = −2 is the solution.

Solving System of Equations by Graphing

In this method, solving system of linear equations is done by plotting their graphs. "The point of intersection of the two lines is the solution of the system of equations using graphical method ."

Example: 3x + 4y = 11 and -x + 2y = 3

Find at least two values of x and y satisfying equation 3x + 4y = 11

So we have 2 points A (1, 2) and B (3, 0.5).

Similarly, find at least two values of x and y satisfying the equation -x + 2y = 3

We have two points C(-3, 0) and D( 3, 3). Plotting these points on the graph we can get the lines in a coordinate plane as shown below.

We observe that the two lines intersect at (1, 2). So, x = 1, y = 2 is the solution of given system of equations. Methods I and II are the algebraic way of solving simultaneous equations and III is the graphical method .

Solving System of Equations Using Cross-multiplication method

In this method, we solve the systems of equations a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0 using the cross-multiplication formula:

x / (b 1 c 2 - b 2 c 1 ) = -y / (a 1 c 2 - a 2 c 1 ) = 1 / (a 1 b 2 - a 2 b 1 )

For more detailed information of this method, click here .

Solving System of Equations Using Matrices

The solution of a system of equations can be solved using matrices . In order to solve a system of equations using matrices, express the given equations in standard form , with the variables and constants on respective sides. for the given equations,

a\(_1\)x + \(b_1\)y + \(c_1\)z = \(d_1\)

a\(_2\)x + \(b_2\)y + \(c_2\)z = \(d_2\)

a\(_3\)x + \(b_3\)y + \(c_3\)z = \(d_3\)

we can express them in the form of matrices as,

\(\left[\begin{array}{ccc} a_1x + b_1y + c_1 z \\ a_2x + b_2y + c_2 z \\ a_3x + b_3y + c_3 z \end{array}\right] = \left[\begin{array}{ccc} d_1 \\ d_2 \\ d_3 \end{array}\right]\)

⇒\(\left[\begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array}\right] + \left[\begin{array}{ccc} x \\ y \\ z \end{array}\right] = \left[\begin{array}{ccc} d_1 \\ d_2 \\ d_3 \end{array}\right]\)

A = \(\left[\begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array}\right]\), X = \(\left[\begin{array}{ccc} x \\ y \\ z \end{array}\right]\), B = \(\left[\begin{array}{ccc} d_1 \\ d_2 \\ d_3 \end{array}\right]\)

⇒ The solution of the system is given by the formula X = A -1 B, where A -1 = inverse of the matrix A. To understand this method more in detail, click here .

Alternatively, Cramer's rule can also be used to solve the system of equations using determinants .

Applications of System of Equations

Systems of equations are a very useful tool and find application in our day-to-day lives for modeling real-life situations and analyzing questions about them.

• System of Equations Word Problems
• Systems of Equations Worksheet

For applying the concept of the system of equations, we need to translate the given situation into two linear equations in two variables , then further solve to find the solution of linear programming problems. Any method to solve the system of equations, substitution, elimination, graphical, etc methods. Follow the below-given steps to apply the system of equations to solve problems in our daily lives,

• To translate and represent the given situation in form of a system of equations, identify unknown quantities in a problem and represent them with variables.
• Write a system of equations modelling the conditions of the problem.
• Solve the system of equations.
• Check and express the obtained solution in terms of the given context.

☛Related Articles:

Check these articles related to the concept of the system of equations.

• Solutions of a Linear Equation
• Simultaneous Linear Equations
• Solving Linear Equations Calculator
• Equation Calculator
• System of Equations Calculator

System of Equations Examples

Example 1: In a ΔLMN, ∠N = 3∠M = 2(∠L+∠M). Calculate the angles of ΔLMN using this information.

Let ∠L = x° and ∠M = y°.

∠N = 3∠M = (3y)°.

∠L + ∠M + ∠N = 180° ∴ x + y + 3y = 180 x + 4y = 180 ... (1) From the given equation, 3∠M = 2(∠L+∠M) ∴ 3y = 2(x + y) ⇒ 2x - y = 0 ... (2) On multiplying (2) by 4 and adding the result to (1), we get 9x = 180 x = 20 Substitute this in (2): 40 - y = 0 ⇒ y = 40 Therefore, ∠L = 20° and ∠M = 40°. Therefore, the third angle is, ∠N = 180 - (20 + 40) = 120°. Answer: ∠L = 20°, ∠M = 40°, and ∠N = 120°.

Example 2: Peter is three times as old as his son. 5 years later, he shall be two and half times as old as his son. What's Peter's present age?

Let Peter's age be x years and his son's age be y years. Then by given info,

x = 3y ... (1)

5 years later,

Peter's age = x + 5 years and his son's age = y + 5 years.

By the given condition,

x + 5 = 5/2 (y + 5)

2x - 5y - 15 = 0 ... (2)

From (1), we have x = 3y. Substitute this in (2):

2(3y) - 5y - 15 = 0

Substitute this in (1): x = 3(15) = 45

Thus, the solution of the given system is (x, y) = (45, 15).

Answer: Present age of Peter = 45 years; Present age of son = 15 years.

Example 3: Tressa starts her job with a certain monthly salary and earns a fixed increment every year. If her salary was $1500 after 4 years and $1800 after 10 years of service, find her starting salary and annual increment.

Let her starting salary be x and annual increment be y.

After 4 years, her salary was $1500

x + 4y = 1500 ... (1)

After 10 years, her salary was $1800

x + 10y = 1800 ... (2)

Subtracting Eqn (1) from (2), we get

On putting y = 50 in (1), we get x = 1300.

Answer: Starting salary was $1300 and annual increment is $50.

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Practice Questions on System of Equations

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FAQs on System of Equations

What is a system of equations in mathematics.

A system of equations in mathematics is a set of linear equations that need to be solved to find a common solution. A real-life problem with two or more unknowns can be converted into a system of equations and can be solved to find a set of values of variables that satisfy all equations.

How Do You Solve a System of Equations?

Solving a system of equations is computing the unknown variables still balancing the equations on both sides. We solve an equation system to find the values of the variables that satisfy the condition of all the given equations. There are different methods to solve a system of equations,

• Graphical method
• Substitution method
• Elimination method

How Do You Create a System of Equations with Two Variables?

To create a system of equations with two variables:

• First, identify the two unknown quantities in the given problem.
• Next, find out the two conditions given and frame equations for each of them.

How to Solve a System of Equations by Substitution Method?

The substitution method is one of the ways to solve a system of equations in two variables, given the set of linear equations. In this method, we substitute the value of a variable found by one equation in the second equation.

How to Solve a System of Equations Using the Elimination Method?

The elimination method is used to solve a system of linear equations. In the elimination method, we eliminate one of the two variables by multiplying each equation with the required numbers and try to solve equations with another variable. In this process, finding LCM of coefficients would be helpful.

What is the Purpose of Graphing Systems of Equations?

To solve the system of equations, given a set of linear equations graphically, we need to find at least two solutions for the respective equations. We observe the pattern of lines after plotting the points to infer it is consistent, dependent, or inconsistent.

• If the two lines are intersecting at the same point, then the intersection point gives a unique solution for the system of equations.
• If the two lines coincide, then in this case there are infinitely many solutions.
• If the two lines are parallel, then in this case there is no solution.

What are Homogeneous System of Linear Equations?

The homogeneous system of linear equations is a set of linear equation each one of which has its constant term to be 0. The process of solving these kind of systems can be learnt in detailed by clicking here .

How to Solve a System of Equations With Cross Multiplication Method?

While solving a system of equations using the cross-multiplication method , we use the formula x / (b 1 c 2 - b 2 c 1 ) = -y / (a 1 c 2 - a 2 c 1 ) = 1 / (a 1 b 2 - a 2 b 1 ) to solve the system a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0.

How do you Solve a System of Equations Using 2 Equations with 3 Variables?

An equation with 3 variables represents a plane .

• Step 1) To solve a system of 2 equations with 3 variables say x, y, and z, we will consider the 1st two equations and eliminate one of the variables, say x, to obtain a new equation.
• Step 2) Next, we write the 2nd variable, y in terms of z from the new equation and substitute it in the third equation.
• Step 3) Assuming z = a, we will obtain values of x and y also in terms of 'a'.
• Step 4) Once, we know the value of a, we can find the values of x and y in terms of 'a'.

What is the Solution to the System of Equations and How to Find It?

Solving system of equations is finding the values of variables that satisfy each equation in that system. There are three main methods to solving system of equations, they are:

Where Can I Find System of Linear Equations Calculator?

The system of linear equations calculator is available here . This allows us to enter the linear equations. Then it will show the solution along with step-by-step solution.

Solving Systems of Equations Using Algebra Calculator

Example problem, how to solve the system of equations in algebra calculator.

• The first equation x+y=7
• Then a comma ,
• Then the second equation x+2y=11

Clickable Demo

More Examples

• Solve y=x+3, y=2x+1: y=x+3, y=2x+1
• Solve 2x+3y=5, x+y=4: 2x+3y=5, x+y=4

Related Articles

• Algebra Calculator Tutorial

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Word Problem: Rachel has 17 apples. She gives some to Sarah. Sarah now has 8 apples. How many apples did Rachel give her?

Simplified Equation: 17 - x = 8

Word Problem: Rhonda has 12 marbles more than Douglas. Douglas has 6 marbles more than Bertha. Rhonda has twice as many marbles as Bertha has. How many marbles does Douglas have?

Variables: Rhonda's marbles is represented by (r), Douglas' marbles is represented by (d) and Bertha's marbles is represented by (b)

Simplified Equation: {r = d + 12, d = b + 6, r = 2 �� b}

Word Problem: if there are 40 cookies all together and Angela takes 10 and Brett takes 5 how many are left?

Simplified: 40 - 10 - 5

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I need a step by step explanation on how you solved the equation. Use substitution to solve the system of equations. 1. x = -2y - 1 4x - 4y = 20

Answer: x=3 and y=−2

Step-by-step explanation:

x=−2y−1;4x−4y=20

Step: Solve x=−2y−1 for x:

Step: Substitute−2y−1 for x in 4x−4y=20:

4(−2y−1)−4y=20

−12y−4=20(Simplify both sides of the equation)

−12y−4+4=20+4(Add 4 to both sides)

(Divide both sides by -12)

Step: Substitute−2 for y x=−2y−1:

x=(−2)(−2)−1

x=3(Simplify both sides of the equation)

Related Questions

At the store, you buy x computer games for $13 each and a magazine for $4. Write an expression in simplest form that represents the total amount of money you spend. asap i will give brianlist

13g + 4m = D

g=number of computer games

m=number of magazines

        hope this work

 If I'm wrong I'm so sorry

If I'm right Thank you (Brainliest plz)

Monica got a $4,500 car loan, to be paid in 3 years. The interest rate is 5%. What will the interest rate is 5%. What will the interest be at the end of 3 years? What is the total she will need to pay back?

QUESTION IS IN PICTURE PLEASE HELP ASAP DONT SKIP

y=4.5x+22 (4.5 goes in first box, 22 goes in second box).

For starters, you can use that line in the middle, a.k.a. the line of best fit, to help set up your equation.

The equation is in slope-intercept (y=mx+b) form, where m is slope and b is y-intercept.

From the given numbers and the graph, it looks like the y-intercept is about 22 so you can drag the 22 in the last box, where the 'b' should be. your equation is now y=mx+22.

For the slope, just calculate rise over run. I've counted and gotten about 9/2, which equals 4.5 (which is in the choices), which is your slope.

Your final equation is now y=4.5x+22. Hope this helps.

Carpet costs $21.95 per square yard and the padding to put under it costs $2.55 per square yard. Felix plans to install padding and carpet in the region shown in the figure. What is the total cost of the carpet and padding needed to exactly cover the room?

$1679.175 or $1679.18

split the carpet into rectangles to make it easier to look at

one rectangle is 2 yards in height and 9 yards in width (5 yards plus 4 yards)

area of a rectangle = (w)(h), so 9 x 2 = 18 yards in area.

second triangle is 4 yards in width, and 3 yards in height.

area of a rectangle = (w)(h), so 4 x 3 = 12 yards in area.

Total area = 30 yards

30(21.95)(2.55) = $1679.175 or $1679.18

Answer: $735

Step-by-step explanation: Hi hawang0604, i still have more questions i need help with.

The graph shows the distance, y, that a car traveled in x hours: A graph is shown with x axis title as Time in hours. The title on the y axis is Distance Traveled in miles. The values on the x axis are from 0 to 5 in increments of 1 for each grid line. The values on the y axis are from 0 to 125 in increments of 25 for each grid line. A line is shown connecting ordered pairs 1, 25 and 2, 50 and 3, 75 and 4, 100. The title of the graph is Rate of Travel. What is the rate of change for the relationship represented in the graph? fraction 1 over 50 fraction 1 over 25 25 50

I would say the answer would be y=25x

Y=35x as in you increase 35 every time you go up on the graph

whats the answer to this question

The thing you have to do is just read the graph

I did it before

If the lowest temperature in Whitesboro for the week was -1°F and the highest temperature was 20°F, what was the total change in temperature? Will give brainliest!!

It went from -1 to 20

it went up or down 21°F

      A        A

     (๑٥〰️٥๑)️                      

ϞϞノ  --  -- 乀  •⃠  

PLS HELP!! Let's say you are borrowing $5000 from the bank to purchase a donkey named Don. The annual (one year) interest rate for the donkey is 4.4%. How much will you be paying in interest for Don?

Which of the following equations shows a proportional function? A y = x + 3 B y = 2x − 1 C y = x − 2 D y = 4x

A function is output = constant •/+/-/÷  input. In any function, the amount of one quantity is dependent on the amount of another.

here, take some dino nuggies.

Did yall eat them all yet?

A park charges $7.50 per hour to rent a bike. The table shows the relationship between the number of hours, h, and the total cost, c. With the information in the table, which expression can be used to find c? A. [tex]\frac{7.50}{}[/tex] B. + 13 C. - 15 D. 7.50

divide both sides by 7.5

C=7.5 hours

Break all sides of 7.5

Which transformation did Benjamin use to create the pattern? rotations translations reflections and rotations reflections and translations

5 ÷ 3/10 Please give me an answer and a answer in fraction form If you can i need an explanation also. Thank you

Please help with a khan academyyy

Help me with this- ILL GIVE BRANILEST

Select the choice that represents: A number increased by 36 equals 52. * A n + 36 B n + 36 = 52 C n - 36 =52 D 36 -n=52

if each dimension of a unit cube is increased by 1. what is the ratio between the surface area of the new cube to that of the original

Original surface area: Each side is 1 * 1 = 1 and there are 6 sides, so 1 * 6 = 6.

New surface area: Each side is 2*2 = 4, and there are 6 sides, so 4 * 6 = 24

The ratio would be 24:6, or 4:1. As a fraction, this would be 4/1

HELPPPPPPPPPPPP PLZZZ

I'm sorry I can't see it, it is blocked.

Answer: yes and then no

Nine years ago, Tarah opened a savings account with her bank. She started with a balance of $359 and has not made any withdrawals or deposits since then. If her interest rate is 8% each year, how much interest has accrued using simple interest? Total Interest = $__________ What is Tarah's new total balance? New Balance Total = $__________

Initial balance in the savings account = $359

Interest rate = 8%

An angle measuring 113° is rotated 180 degrees clockwise. What is the measure of the angle after the rotation?

Answer: 293 degrees

113+180=293 degrees

293 i took the test its correct

You decide to open a small business in the Bronx. You borrow $15,000 from the bank at an annual interest rate of 7% for 24 months. What is the amount of interest you will pay on this loan?

$2,100 interest

15,000×7% = 1,050

1,050×2 = 2,100

15000x2x7/100 = 210

can someone help me with this problem y≤ 1±10

(ZOOM IN TO SEE THE QUESTION) I need help!!!

359 is your answer

Have a good day ;-)

To pay for a concert ticket, Geoff needs to earn $135 in commission. What must Geoff’s sales total for the week?

A map has scale of 1.5 inches =9 miles. On the map, two towns are 4 inches apart. What distance, in miles, between the two town?

the answer is 24 miles.

please help i need help really bad

Emmy read a total of 28 books over 4 months. After belonging to the book club for 7 months, how many books will Emmy have read in all? Assume the relationship is directly proportional.

A survey asked 200 Grade 4 students about their reading preferences. Based on the Graph, how many of them would you expect to prefer comic books? A. 10 B. 20 C. 60 D. 80 E. 100

Based on the Graph , 10 would you expect to prefer comic books. So, the correct option is A .

A graph is a framework made up of a number of things, some of which are paired together to form "related" objects . The items are represented by mathematical abstractions known as vertices, and each pair of connected vertices is referred to as an edge .

The link between arcs and points is described by a graph, which is a mathematical description of a network. A graph is made up of certain locations and the connecting lines. It doesn't matter how long the lines are or where the points are located. A node is the name for each element in a graph. Based on the given Graph , 10 would you expect to prefer comic books.

Therefore, the correct option is A .

Learn more about Graph , here:

https://brainly.com/question/17267403

45% of ____ is 396 what is the blank?

Explanation:

CAN SOME ONE ANSWER MY QUESTION I NEED HELP you have a shelf that holds 15 1/2 pounds how many 1 1/4 pound books can the shelf hold

15 1/2=31/2

31/2÷5/4=31/2×4/5

31/2×4/5=124/10

124/10=12.4

Because books cannot come as parts, we round this down to 12.

How to solve math equations in any app on iPhone, iPad, or Mac

Learn how to solve basic mathematical expressions and do calculations directly inside iPhone, iPad, and Mac apps such as Notes, Messages, Mail, and more.

Up until now, when you had to perform a calculation, you’d obviously open your iPhone’s Calculator app. If you were on an iPad, you had to rely on some third-party solution like the excellent Solves app .

iOS 18, iPadOS 18, and macOS Sequoia introduce system-wide math solving abilities. Surely, it isn’t as powerful as Apple’s dedicated Calculator app, but it gets the work done without you having to leave the app you are in.

Note that iPadOS 18 finally adds a powerful Calculator app to the iPad .

Apps that support math calculations

While you technically can’t do math operations in all places, the list of apps where it works in iOS 18 beta is comprehensive.

I tried all Apple apps on my iPhone running iOS 18 developer beta, and as of now, you can perform basic calculations in the following apps. Note that they should also work on iPads running iPadOS 18.

• Messages: Inside the text field when typing a text.
• Notes: Inside a note.
• Mail: When composing an email.
• Calendar: When typing the title, location, and notes while creating a new event or reminder .
• Reminders: When typing the task name or note below it.
• Freeform: In a text box.
• Journal: When you are making a diary entry .
• Translate: When typing in the translation box.
• Spotlight: Enter a math problem in the search bar, and you will see the answer. Note that this works even in older versions of iOS.
• Safari: Similar to Spotlight, you can enter a simple math query in your browser’s address bar to get its answer. This, too, works in older versions of iOS.

Surprisingly, math calculations didn’t work in Apple’s iWork suite of apps: Pages, Keynote, and Numbers. I imagine an app update will add it.

I also tried some third-party apps like WhatsApp, but it didn’t work. Hopefully, Apple allows third-party developers to add this feature to their apps.

Do quick calculations anywhere on your iPhone and iPad

• Open a supported app like Messages or Notes. We are using Notes.
• Go inside a note or create one.
• Type what you want to calculate. I’m going with 45*34/5.
• Add the equal to sign (=) , and now you will see the answer to that math problem above your keyboard and also next to what you typed.
• Hit the return key on your keyboard or tap the middle suggestion over the keyboard to insert that answer to your note.

To sum up: System-wide math calculations work in most stock apps. Just enter the values in a text field and add the equals sign to see the answer. Now, tap the return key or tap the problem along with its answer that appears above the on-screen keyboard.

Important: You can also add a currency symbol (like $), and your phone will figure out the answer accordingly.

If you use an Apple Pencil

Use Scribble to handwrite your math equation in an app like Notes or in a text field such as Messages. Remember to draw the equal to sign (=). iPadOS 18 will understand your handwriting and display the right answer.

Number Line

• x+y+z=25,\:5x+3y+2z=0,\:y-z=6
• x+2y=2x-5,\:x-y=3
• 5x+3y=7,\:3x-5y=-23
• x+z=1,\:x+2z=4

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• High School Math Solutions – Systems of Equations Calculator, Nonlinear In a previous post, we learned about how to solve a system of linear equations. In this post, we will learn how...

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Bifurcation analysis, modulation instability and optical soliton solutions and their wave propagation insights to the variable coefficient nonlinear Schrödinger equation with Kerr law nonlinearity

• Published: 02 July 2024

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• Md. Mamunur Roshid 1 , 2 &
• M. M. Rahman 1  

The present paper aims to explore bifurcation analysis, modulation instability, and optical soliton solutions in nonlinear media with third-order dispersion terms. A variable coefficient third-order nonlinear Schrödinger’s equation (PNLSE) with truncated \(M\) -fractional derivative is considered. We also discuss a few assets that the derivative satisfies. Initially, a novel evaluation method known as bifurcation analysis is employed to look at the complex model's dynamic behavior. Figure  1 provides an analytical and graphical study of the observed mechanism of static soliton through a saddle-node bifurcation in the nonlinear Schrödinger problem using a matching technique. Secondly, By implementing novel two analytic techniques such as unified solver and generalized unified techniques to offer insights into wave propagation and optical soliton conduct in nonlinear optics, optical communications, quantum mechanics, plasma physics, and engineering. The underlying idea of these two approaches is to transform the equation with partial derivatives into a version of the equation with ordinary derivatives, which are first required to incorporate a new wave definition. These techniques permit us to generate innovative soliton solutions that can be formulated in terms of rational, hyperbolic, and trigonometrical functions. The collected outcomes have the potential to facilitate an understanding and elucidation of the physical characteristics of waves moving within a dispersive substance. We additionally estimated conservative values related to solitons, such as energy, momentum, and power. By selecting appropriate parameter values, the graphical shapes in 3D, density, and 2D are generated using relevant parameter values to visually present the obtained results, including periodic wave soliton, interaction of kink and bell shape soliton, periodic breather wave soliton, double periodic soliton through unified solver method and interaction of kink and periodic lump wave soliton and also with the periodic soliton, double periodic wave soliton, periodic wave soliton with lump soliton, multi periodic wave with kink shape soliton, periodic wave soliton, periodic breather wave soliton, periodic wave soliton, multi periodic wave soliton, periodic lump wave soliton, breather wave soliton multi periodic wave through generalized unified technique, providing an insightful visualization of the discovered solutions. In a two-dimensional graph, we show the effect of truncated \(M\) -fractional parameters for [ \(g=0.1, 0.5, 0.9\) ]. Additionally, the modulation instability spectrum can be expressed utilizing a linear analysis technique, and the modulation instability bands are shown to be influenced by the third-order and group velocity dispersion. The findings indicate that the modulation instability disappears for negative values of the fourth order in a typical dispersion regime. Consequently, it was shown that the techniques mentioned previously could be an effective tool to generate unique, precise soliton solutions for numerous uses, which are crucial to nonlinear optics, optical communications, and engineering.

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Osman, M.S., Machado, J.A.T., Baleanu, D., Zafar, A., Raheel, M.: On distinctive solitons type solutions for some important nonlinear Schrödinger equations. Opt. Quantum Electron. 53 (2), 70 (2021)

Article   Google Scholar  

Wazwaz, A.M.: A variety of multiple-soliton solutions for the integrable (4+ 1)-dimensional Fokas equation. Waves Random Complex Media. 31 (1), 46–56 (2020)

Article   MathSciNet   Google Scholar  

Lü, X., Ma, W.X., Yu, J., Lin, F., Khalique, C.M.: Envelope bright-and dark-soliton solutions for the Gerdjikov-Ivanov model. Nonlinear Dyn. 82 , 1211–1220 (2015)

Ali, A., Ahmad, J., Javed, S., Hussain, R., Alaoui, M.K.: Numerical simulation and investigation of soliton solutions and chaotic behavior to a stochastic nonlinear Schrödinger model with a random potential. PLoS ONE 19 (1), e0296678 (2024)

Arnous, A.H., Nofal, T.A., Biswas, A., Yıldırım, Y., Asiri, A.: Cubic-quartic optical solitons of the complex Ginzburg-Landau equation: a novel approach. Nonlinear Dyn. 111 , 1–16 (2023)

Asghari, Y., Eslami, M., Matinfar, M., Rezazadeh, H.: Novel soliton solution of discrete nonlinear Schrödinger system in nonlinear optical fiber. Alex. Eng. J. 90 , 7–16 (2024)

Ghanbari, B., Inc, M.: A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear Schrödinger equation. Eur. Phys. J. Plus. 133 (4), 142 (2018)

Kalita, J., Das, R., Hosseini, K., Baleanu, D., Salahshour, S.: Solitons in magnetized plasma with electron inertia under weakly relativistic effect. Nonlinear Dyn. 111 (4), 3701–3711 (2023)

Akram, G., Sadaf, M., Khan, M.A.U.: Soliton solutions of the resonant nonlinear Schrödinger equation using modified auxiliary equation method with three different nonlinearities. Math. Comput. Simul 206 (4), 1–20 (2023)

Google Scholar  

Asghari, Y., Eslami, M., Rezazadeh, H.: Soliton solutions for the time-fractional nonlinear differential-difference equation with conformable derivatives in the ferroelectric materials. Opt. Quantum Electron. 55 , 289 (2023)

Zahran, E.H., Khater, M.M.: Modified extended tanh-function method and its applications to the Bogoyavlenskii equation. Appl. Math. Model. 40 (3), 1769–1775 (2016)

Roshid, M.M., Rahman, M.M., Roshid, H.O.: Effect of the nonlinear dispersive coefficient on time-dependent variable coefficient soliton solutions of Kolmogorov–Petrovsky–Piskunov arising in biological and chemical science. Heliyon. (2024). https://doi.org/10.1016/j.heliyon.2024.e31294

Hossain, S., Roshid, M.M., Uddin, M., Ripa, A.A., Roshid, H.O.: Abundant time-wavering solutions of a modified regularized long wave model using the EMSE technique. Partial Differ. Equ. Appl. Math. 8 , 100551 (2023)

Osman, M.S., Ghanbari, B.: New optical solitary wave solutions of Fokas-Lenells equation in presence of perturbation terms by a novel approach. Optik 175 , 328–333 (2018)

Ma, W.X., Lee, J.H.: A transformed rational function method and exact solutions to the 3+ 1 dimensional Jimbo-Miwa equation. Chaos Solit. Fractals. 42 (3), 1356–1363 (2009)

Raza, N., Osman, M.S., Abdel-Aty, A.H., Abdel-Khalek, S., Besbes, H.R.: Optical solitons of space-time fractional Fokas-Lenells equation with two versatile integration architectures. Adv. Differ. Equ. 2020 , 1–15 (2020)

Younas, U., Yao, F., Nasreen, N., Khan, A., Abdeljawad, T.: Dynamics of M-truncated optical solitons and other solutions to the fractional Kudryashov’s equation. Results Phys. 58 , 107503 (2024)

Younas, U., Yao, F., Ismael, H.F., Sulaiman, T.A., Murad, M.A.: Sensitivity analysis and propagation of optical solitons in dual-core fiber optics. Opt. Quantum Electron. 56 (4), 548 (2024). https://doi.org/10.1007/s11082-023-06220-7

Younas, U., Yao, F., Nasreen, N., Khan, A., Abdeljawad, T.: On the dynamics of soliton solutions for the nonlinear fractional dynamical system: application in ultrasound imaging. Results Phys. 57 , 107349 (2024)

Younas, U., Ismael, H.F., Sulaiman, T.A.: Dynamics of M-truncated optical solitons in fiber optics governed by fractional dynamical system. Opt. Quant. Electron. 56 , 25 (2024)

Ma, W.X.: A refined invariant subspace method and applications to evolution equations. Sci China Math 55 , 1769–1778 (2012)

Roshid, M.M., Abdeljabbar, A., Aldurayhim, A., Rahman, M.M.: Dynamical interaction of solitary, periodic, rogue type wave solutions and multi-soliton solutions of the nonlinear models. Heliyon. 8 (12), e11996 (2022)

Roshid, M.M., Rahman, M.M., Bashar, M.H., Hossain, M.M., Mannaf, M.A.: Dynamical simulation of wave solutions for the M-fractional Lonngren-wave equation using two distinct methods. Alex. Eng. J. 81 , 460–468 (2023)

Wazwaz, A.M., Kaur, L.: New integrable Boussinesq equations of distinct dimensions with diverse variety of soliton solutions. Nonlinear Dyn. 97 , 83–94 (2019)

Rehman, H.U., Said, G.S., Amer, A., Ashraf, H., Tharwat, M.M., Abdel-Aty, M., Elazab, N.S., Osman, M.S.: Unraveling the (4+ 1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili equation: exploring soliton solutions via multiple techniques. Alex. Eng. J. 90 , 17–23 (2024)

Roshid, M.M., Hossain, M.M., Hasan, M.S., Munshi, M.J.H., Sajib, A.H.: Dynamical structure of truncated M− fractional Klein-Gordon model via two integral schemes. Results Phys. 46 , 106272 (2023)

Priyanka, S., Arora, F., Mebrek-Oudina, S.: Sahani, Super convergence analysis of fully discrete Hermite splines to simulate wave behaviour of Kuramoto-Sivashinsky equation. Wave Motion 121 , 103187 (2023)

Farhan, M., Omar, Z., Mebarek-Oudina, F., Raza, J., Shah, Z., Choudhari, R.V., Makinde, O.D.: Implementation of the one-step one-hybrid block method on the nonlinear equation of a circular sector oscillator. Comput. Math. Model. 31 (1), 116–132 (2020). https://doi.org/10.1007/s10598-020-09480-0

Osman, M.S., Ali, K.K., Gómez-Aguilar, J.F.: A variety of new optical soliton solutions related to the nonlinear Schrödinger equation with time-dependent coefficients. Optik 222 , 165389 (2020)

Bilal, M., Rehaman, S.-U., Ahmad, J.: Dispersive solitary wave solutions for the dynamical soliton model by three versatile analytical mathematical methods. Eur. Phys. J. Plus 137 , 674 (2022)

Bilal, M., Younas, U., Ren, J.: Dynamics of exact soliton solutions to the coupled nonlinear system using reliable analytical mathematical approaches. Commun. Theor. Phys. 73 , 085005 (2021)

Bilal, M., Ahmad, J.: A variety of exact optical soliton solutions to the generalized (2+1)-dimensional dynamical conformable fractional Schrödinger model. Results Phys. 33 , 105198 (2022)

Manikandan, K., Serikbayev, N., Aravinthan, D., Hosseini, K.: Solitary wave solutions of the conformable space–time fractional coupled diffusion equation. Partial Differ. Equ. Appl. Math. 9 , 100630 (2024)

Wang, B.H., Wang, Y.Y., Dai, C.Q., Chen, Y.X.: Dynamical characteristic of analytical fractional solitons for the space–time fractional fokas-lenells equation. Alex. Eng. J. 59 , 4699–4707 (2020)

Ameen, I.G., Elboree, M.K., Taie, R.O.A.: Traveling wave solutions to the nonlinear space–time fractional extended KdV equation via efficient analytical approaches. Alex. Eng. J. 82 , 468–483 (2023)

Umer, A., Abbas, M., Shafiq, M., Abdullah, F.A., Sen, M.D.I., Abdeljawad, T.: Numerical solutions of Atangana-Baleanu time-fractional advection diffusion equation via an extended cubic B-spline technique. Alex. Eng. J. 74 , 285–300 (2023)

Atangana, A., Alqahtani, R.T.: Modelling the spread of river blindness disease via the Caputo fractional derivative and the beta-derivative. Entropy 18 (2), 40 (2016)

Zainab, I., Akram, G.: Effect of β-derivative on time fractional Jaulent-Miodek system under modified auxiliary equation method and exp (-g(Ω))- expansion method. Chaos Solit. Fractals. 168 , 113147 (2023)

Akram, G., Sadaf, M., Zainab, I.: Observations of fractional effects of β-derivative and M-truncated derivative for space time fractional Phi-4 equation via two analytical techniques. Chaos Solit. Fractals. 154 , 111645 (2022)

Yao, S.W., Manzoor, R., Zafar, A., Inc, M., Abbagari, S., Houwe, A.: Exact soliton solutions to the Cahn-Allen equation and Predator-Prey model with truncated M-fractional derivative. Results Phys. 37 , 105455 (2022)

Sadaf, M., Akram, G., Arshed, S., Farooq, K.: A study of fractional complex Ginzburg-Landau model with three kinds of fractional operators. Chaos Solit. Fractal. 166 , 112976 (2023)

Zafar, A., Ashraf, M., Saboor, A., Bekir, A.: M-Fractional soliton solutions of fifth order generalized nonlinear fractional differential equation via (G′/G2)-expansion method. Phys. Scr. 99 , 025242 (2024)

Taghizadeh, N., Mirzazadeh, M.: The simplest equation method to study perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity. Commun. Nonlinear Sci. Numer. Simulat. 17 , 1493–1499 (2012)

Ma, W.X., Chen, M.: Direct search for exact solutions to the nonlinear Schrödinger equation. Appl. Math. Comput. 215 , 2835–2842 (2009)

MathSciNet   Google Scholar  

Sweilam, N.H.: Variational iteration method for solving cubic nonlinear Schrödinger equation. J. Comput. Appl. Math. 207 (1), 155–163 (2007)

Taghizadeh, N., Mirzazadeh, M., Farahrooz, F.: Exact solutions of the nonlinear Schrödinger equation by the first integral method. J. Math. Anal. Appl. 374 (2), 549–553 (2011)

Zhou, Y., Wang, M., Miao, T.: The periodic wave solutions and solitary for a class of nonlinear partial differential equations. Phys. Lett. A 323 , 77–88 (2004)

Forestieri, E., Secondini, M.: Solving the nonlinear schrödinger equation. Opt. Commun. Theory Techn. 2005 , 3–11 (2005)

Boulaaras, S.M., Rehman, H.U., Iqbal, I., Sallah, M., Qayyum, A.: Unveiling optical solitons: solving two forms of nonlinear Schrödinger equations with unified solver method. Optik 295 , 171535 (2023)

Alkhidhr, H.A., Abdelrahman, M.A.: Wave structures to the three coupled nonlinear Maccari’s systems in plasma physics. Results Phys. 33 , 105092 (2022)

Turgut, A.K., Tugba, S.A., Saha, A., Kara, A.H.: Propagation of nonlinear shock waves for the generalized Oskolkov equation and its dynamic motions in the presence of an external periodic perturbation. Pramana J. Phys. 90 , 78 (2018)

Roshid, M.M., Uddin, M., Mostafa, G.: Dynamic optical soliton solutions for M-fractional Paraxial Wave equation using unified technique. Result Phys. 51 , 106632 (2023)

Roshid, M.M., Rahman, M.M., Roshid, H.-O., Bashar, M.H.: A variety of soliton solutions of time M-fractional non-linear models via a unified technique. PLoS ONE 19 (4), e0300321 (2024)

Bilal, M., Hu, W., Ren, J.: Different wave structures to the Chen–Lee–Liu equation of monomode fibers and its modulation instability analysis. Eur. Phys. J. Plus 136 , 385 (2021)

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Acknowledgements

The authors like to express their gratitude to the Research Grant (Grant No. 1111202309017). Bangladesh University of Engineering and Technology (BUET)

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Roshid, M.M., Rahman, M.M. Bifurcation analysis, modulation instability and optical soliton solutions and their wave propagation insights to the variable coefficient nonlinear Schrödinger equation with Kerr law nonlinearity. Nonlinear Dyn (2024). https://doi.org/10.1007/s11071-024-09872-6

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DOI : https://doi.org/10.1007/s11071-024-09872-6

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Numerical solutions of potential flow equations using finite differences, lucy annastacia, collins andete, collins wanyama, benjamin mwendwa, griffin omondi.

This article delves into the numerical solutions of potential flow equations using finite differences, aiming to enhance our understanding  of fluid dynamics. The general objective is to obtain numerical solutions to potential flow equations using finite differences, with specific  objectives including the investigation of potential flow equations, the solutions of associated PDE and the analysis of the stability of  employed numerical schemes. The study employs a combination of numerical methods to achieve its objectives; MATLAB is utilized as a  computational tool, while the Gauss-Seidel and Jacobi’s iterative methods are implemented for solving PDEs. Central differences are  employed for discretization. The study yields valuable insights into the behaviour of potential flow systems. The significance of this  research lies in its contribution to advancing our comprehension of fluid dynamics with potential applications. Generally, this work  provides a foundation for further exploration and application of numerical methods in the study of potential flow.

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