simultaneous equations problem solving tes

Solving Simultaneous Equations: Lessons

If you want a complete lesson, a Tarsia jigsaw, or a fun and engaging lesson activity, then you have come to the right place!

Complete Lesson PowerPoints keyboard_arrow_up Back to Top

Complete lessons from some of my favourite authors. There are few better places to your own planning process.

Tarsia Jigsaws and Card Sorts keyboard_arrow_up Back to Top

I have been a Tarsia fan for many years. Visit the my tarsia page for lots of different ideas for using this wonderful free resource, and to get hold of the software yourself.

Lesson Activities keyboard_arrow_up Back to Top

A selection of some of my favourite, free maths activities to use in lessons. I have tried out each and every one with my students.

Solving Simultaneous Equations

An introduction to simultaneous equations using the bar model, and with no scaling up. The questions are from Minimally Different Questions , which is definitely worth exploring.

Linear simultaneous equations starts with visual questions involving burgers and chips, and moves towards the algebraic method. Main task is differentiated and answers are included.

Solving linear simultaneous equations graphically. Make sure students can sketch linear graphs first.

Nonlinear simultaneous equations includes visual examples of solving, as well as algebraic. Main task is differentiated and answers are included.

Simultaneous Equations

Simultaneous equations are two or more algebraic equations that share common variables and are solved at the same time (that is, simultaneously). For example, equations x + y = 5 and x - y = 6 are simultaneous equations as they have the same unknown variables x and y and are solved simultaneously to determine the value of the variables. We can solve simultaneous equations using different methods such as substitution method, elimination method, and graphically.

In this article, we will explore the concept of simultaneous equations and learn how to solve them using different methods of solving. We shall discuss the simultaneous equations rules and also solve a few examples based on the concept for a better understanding.

What are Simultaneous Equations?

Simultaneous equations are two or more algebraic equations with the same unknown variables and the same value of the variables satisfies all such equations. This implies that the simultaneous equations have a common solution. Some of the examples of simultaneous equations are:

• 2x - 4y = 4, 5x + 8y = 3
• 2a - 3b + c = 9, a + b + c = 2, a - b - c = 9
• 3x - y = 5, x - y = 4
• a 2 + b 2 = 9, a 2 - b 2 = 16

We can solve such a set of equations using different methods. Let us discuss different methods to solve simultaneous equations in the next section.

Solving Simultaneous Equations

We use different methods to solve simultaneous equations. Some of the common methods are:

• Substitution Method
• Elimination Method
• Graphical Method

Simultaneous equations can have no solution, an infinite number of solutions, or unique solutions depending upon the coefficients of the variables. We can also use the method of cross multiplication and determinant method to solve linear simultaneous equations in two variables . We can add/subtract the equations depending upon the sign of the coefficients of the variables to solve them.

To solve simultaneous equations, we need the same number of equations as the number of unknown variables involved. We shall discuss each of these methods in detail in the upcoming sections with examples to understand their applications properly.

Simultaneous Equations Rules

To solve simultaneous equations, we follow certain rules first to simplify the equations. Some of the important rules are:

• Simplify each side of the equation first by removing the parentheses, if any.
• Combine the  like terms .
• Isolate the variable terms on one side of the equation.
• Then, use the appropriate method to solve for the variable.

Solving Simultaneous Equations Using Substitution Method

Now that we have discussed different methods to solve simultaneous equations. Let us solve a few examples using the substitution method to understand it better. Consider a system of equations x + y = 4 and 2x - 3y = 9. Now, we will find the value of one variable in terms of another variable using one of the equations and substitute it into the other equation. We have

x + y = 4 --- (1)

2x - 3y = 9 --- (2)

From (1), we have

x = 4 - y --- (3)

Substituting this in (2), we get

2(4 - y) - 3y = 9

⇒ 8 - 2y - 3y = 9

⇒ 8 - 5y = 9

Isolating the variable term to one side of the equation, we have

⇒ -5y = 9 - 8

⇒ y = 1/(-5)

Substituting the value of y in (3), we have

x = 4 - (-1/5)

= (20 + 1)/5

Answer: So, the solution of the simultaneous equations x + y = 4 and 2x - 3y = 9 is x = 21/5 and y = -1/5.

Solving Simultaneous Equations By Elimination Method

To solve simultaneous equations by the elimination method, we eliminate a variable from one equation using another to find the value of the other variable. Let us solve an example to understand find the solution of simultaneous equations using the elimination method. Consider equations 2x - 5y = 3 and 3x - 2y = 5. We have

2x - 5y = 3 --- (1)

and 3x - 2y = 5 --- (2)

Here, we will eliminate the variable y, so we find the LCM of the coefficients of y. LCM (5, 2) = 10. So, multiply equation (1) by 2 and equation (2) by 5. So, we have

[ 2x - 5y = 3 ] × 2

⇒ 4x - 10y = 6 --- (3)

[ 3x - 2y = 5 ] × 5

⇒ 15x - 10y = 25 --- (4)

Now, subtracting equation (3) from (4), we have

(15x - 10y) - (4x - 10y) = 25 - 6

⇒ 15x - 10y - 4x + 10y = 19

⇒ (15x - 4x) + (-10y + 10y) = 19

⇒ 11x + 0 = 19

⇒ x = 19/11

Now, substituting this value of x in (1), we have

2(19/11) - 5y = 3

⇒ 38/11 - 5y = 3

⇒ 5y = 38/11 - 3

⇒ 5y = (38 - 33) / 11

⇒ y = 5/(11×5)

So, the solution of the simultaneous equations 2x - 5y = 3 and 3x - 2y = 5 using the elimination method is x = 19/11 and y = 1/11.

Solving Simultaneous Equations Graphically

In this section, we will learn to solve the simultaneous equations using the graphical method. We will plot the lines on the coordinate plane and then find the point of intersection of the lines to find the solution. Consider simultaneous equations x + y = 10 and x - y = 4. Now, find two points (x, y) satisfying for each equation such that the equation holds.

For x + y = 10, we have

So, we have coordinates (0, 10) and (10, 0). Plot them and join the points and plot the line x + y = 10.

For equation x - y = 4, we have

So, we have coordinates (0, -4) and (4, 0). Plot them and join the points and plot the line x - y = 4.

Now, as we have plotted the two lines, find their intersecting point. The two lines x + y = 10 and x - y = 4 intersect each other at (7, 3). So, we have found the solution of the simultaneous equations x + y = 10 and x - y = 4 graphically which is x = 7 and y = 3.

Important Notes on Simultaneous Equations

• Simultaneous equations are two or more algebraic equations that share common variables and are solved at the same time.
• Simultaneous equations can be solved using different methods such as substitution method, elimination method, and graphically.
• We can also use the cross multiplication and determinant method to solve simultaneous linear equations in two variables.

☛ Related Articles:

• Solutions of a Linear Equation
• Simultaneous Linear Equations

Simultaneous Equations Examples

Example 1: Solve the simultaneous equations 2x - y = 5 and y - 4x = 1 using the appropriate method.

Solution: To solve 2x - y = 5 and y - 4x = 1, we will use the elimination method as it is easy to eliminate the variable y by adding the two equations. So, we have

2x - y = 5 --- (1)

y - 4x = 1 --- (2)

Adding (1) and (2), we get

(2x - y) + (y - 4x) = 5 + 1

⇒ 2x - y + y - 4x = 6

Substitute this value of x in (1)

2(-3) - y = 5

⇒ -6 - y = 5

⇒ y = -6 - 5

Answer: Solution of simultaneous equations 2x - y = 5 and y - 4x = 1 is x = -3 and y = -11.

Example 2: Find the solution of the simultaneous equations 2x - 4y + z = 2, x + 5y - 3z = 7, 3x + 2y - z = 10 using the substitution method.

Solution: We have

2x - 4y + z = 2 --- (1)

x + 5y - 3z = 7 --- (2)

3x + 2y - z = 10 --- (3)

z = 2 - 2x + 4y

Substituting this value of z in (2) and (3),

x + 5y - 3(2 - 2x + 4y) = 7

⇒ x + 5y - 6 + 6x - 12y = 7

⇒ 7x - 7y = 13 --- (4)

3x + 2y - (2 - 2x + 4y) = 10

3x + 2y - 2 + 2x - 4y = 10

⇒ 5x - 2y = 12 --- (5)

Now, solving the two-variable equations (4) and (5), multiply (4) by 2 and (5) by 7, we have

[7x - 7y = 13 ] × 2 and [5x - 2y = 12 ] × 7

⇒ 14x - 14y = 26 and 35x - 14y = 84

Now, subtracting the above two equations, we have

(14x - 14y) - (35x - 14y)= 26 - 84

⇒ 14x - 35x - 14y + 14y = -58

⇒ -21x = -58

⇒ x = 58/21 --- (A)

Substitute the value of x in (5)

5(58/21) - 2y = 12

⇒ 290/21 - 2y = 12

⇒ 2y = 290/21 - 12

= (290 - 252)/21

⇒ y = 19/21 --- (B)

Substituting the values of x and y in z = 2 - 2x + 4y, we have

z = 2 - 2(58/21) + 4(19/21)

= (42 - 116 + 76)/21

= 2/21 --- (C)

From (A), (B), (C), we have x = 58/21, y = 19/21, and z = 2/21

Answer: Solution is x = 58/21, y = 19/21, and z = 2/21.

Example 3: Find the solution of simultaneous equations x - y = 10 and 2x + y = 9.

Solution: We will solve the given equations using the elimination method.

Adding x - y = 10 and 2x + y = 9, we have

(x - y) + (2x + y) = 10 + 9

⇒ x + 2x - y + y = 19

So, we have

19/3 - y = 10

⇒ y = 19/3 - 10

= (19 - 30)/3

Answer: The solution of x - y = 10 and 2x + y = 9 is x = 19/3 and y = -11/3.

go to slide go to slide go to slide

Book a Free Trial Class

Simultaneous Equations Questions

go to slide go to slide

FAQs on Simultaneous Equations

Simultaneous equations are two or more algebraic equations that share common variables and are solved at the same time (that is, simultaneously).

How to Solve Simultaneous Equations?

What is the substitution method in simultaneous equations.

According to the substitution method, we obtain the value of one variable in terms of another and then substitute that into another equation to find the value of the other variable.

What is the Rule for Simultaneous Equations?

Some of the important rules of simultaneous equations are:

• Combine the like terms.

What are Linear Simultaneous Equations?

Linear simultaneous equations refer to simultaneous equations where the degree of the variables is one.

How to Solve 3 Simultaneous Equations?

We can solve 3 simultaneous equations using various methods such as:

It also depends upon the number of variables involved.

What are the Three Methods to Solve Simultaneous Equations?

The three methods to solve simultaneous equations are:

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Unit 1: Linear simultaneous equations

Geometrical representation.

• Solutions to systems of equations: dependent vs. independent (Opens a modal)
• Number of solutions to a system of equations (Opens a modal)

Solution of simultaneous equations by use of graphs

• Systems of equations with graphing: 5x+3y=7 & 3x-2y=8 (Opens a modal)
• Number of solutions to a system of equations (graphically) Get 3 of 4 questions to level up!

Conditions of solvability of two linear simultaneous equations

• No videos or articles available in this lesson
• Number of solutions of system of equations Get 3 of 4 questions to level up!
• Number of solutions to systems of equations (intermediate) Get 3 of 4 questions to level up!

Method of substitution

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

Method of elimination

• Systems of equations with elimination (and manipulation) (Opens a modal)
• Solving systems of equations by elimination (old) (Opens a modal)
• 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!

Cross multiplication

• Solving system of equations through cross multiplication Get 3 of 4 questions to level up!

Equations reducible to linear form

• Identifying substitutions: Equations reducible to linear form Get 3 of 4 questions to level up!
• Solving equations reducible to linear form Get 3 of 4 questions to level up!
• Word problems: Writing equations reducible to linear form Get 3 of 4 questions to level up!

Linear equations word problems

• Age word problem: Ben & William (Opens a modal)
• Forming equations with two variables Get 3 of 4 questions to level up!
• Age word problems Get 3 of 4 questions to level up!
• Word problems involving pair of linear equations (advanced) Get 3 of 4 questions to level up!
• Systems of equations word problems Get 3 of 4 questions to level up!

Simultaneous Equations True or False ( Editable Word | PDF | Answers )

Solving Simultaneous Equations (Same y Coefficients) Fill in the Blanks ( Editable Word | PDF | Answers )

Solving Simultaneous Equations (Same y Coefficients) Practice Strips ( Editable Word | PDF |  Answers )

Solving Simultaneous Equations (Same x Coefficients) Fill in the Blanks ( Editable Word | PDF | Answers )

Solving Simultaneous Equations (Different y Coefficients) Fill in the Blanks ( Editable Word | PDF | Answers )

Solving Simultaneous Equations (Different y Coefficients) Practice Strips ( Editable Word | PDF | Answers )

Solving Simultaneous Equations (Different x Coefficients) Fill in the Blanks ( Editable Word | PDF | Answers )

Solving Simultaneous Equations Sort It Out ( Editable Word | PDF | Answers )

Linear Simultaneous Equations Crack the Code ( Editable Word | PDF | Answers )

Linear Simultaneous Equations Worded Problems Practice Strips ( Editable Word | PDF | Answers )

Worded Simultaneous Equations Name the Film ( Editable Word | PDF | Answers ​ )

Linear Simultaneous Equations Revision Practice Grid ( Editable Word | PDF | Answers )

Investigating Linear Simultaneous Equations and Graphs Activity ( Editable Word | PDF | Answers )

Solving Linear Simultaneous Equations Graphically Practice Grid ( Editable Word | PDF | Answers )

Solving Linear Simultaneous Equations by Substitution Practice Strips ( Editable Word | PDF | Answers )

Solving Non-Linear Simultaneous Equations Fill in the Blanks ( Editable Word | PDF | Answers )

Non-Linear Simultaneous Equations Practice Strips ( Editable Word | PDF | Answers )

Harder Simultaneous Equations Practice Grid ( Editable Word | PDF | Answers )

• International
• Schools directory
• Resources Jobs Schools directory News Search

Simultaneous Equations

Subject: Mathematics

Age range: 14-16

Resource type: Worksheet/Activity

Last updated

5 December 2023

• Share through email
• Share through twitter
• Share through linkedin
• Share through facebook
• Share through pinterest

This product consists of 3 worksheets on the following topics:

• Linear Simultaneous Equations (Easier Questions),
• Linear Simultaneous Equations (More Difficult Questions), and
• Linear and Quadratic Simultaneous Equations.

Questions increase in sophistication through the use of fluency as well as reasoning and problem-solving sections.

Each worksheet consists of one A4 page with questions. Each worksheet comes with answers.

Created by teachers for teachers. Great for KS3 and KS4 students.

Tes paid licence How can I reuse this?

Your rating is required to reflect your happiness.

It's good to leave some feedback.

Something went wrong, please try again later.

This resource hasn't been reviewed yet

To ensure quality for our reviews, only customers who have purchased this resource can review it

Report this resource to let us know if it violates our terms and conditions. Our customer service team will review your report and will be in touch.

Not quite what you were looking for? Search by keyword to find the right resource:

All About Maths

Create, edit, and share assessments quickly and easily with Exampro Mathematics

Quick links

• AQA exams and results services

Registration Required