how to solve algebraic equations with i

• PRO Courses Guides New Tech Help Pro Expert Videos About wikiHow Pro Upgrade Sign In
• EDIT Edit this Article
• EXPLORE Tech Help Pro About Us Random Article Quizzes Request a New Article Community Dashboard This Or That Game Popular Categories Arts and Entertainment Artwork Books Movies Computers and Electronics Computers Phone Skills Technology Hacks Health Men's Health Mental Health Women's Health Relationships Dating Love Relationship Issues Hobbies and Crafts Crafts Drawing Games Education & Communication Communication Skills Personal Development Studying Personal Care and Style Fashion Hair Care Personal Hygiene Youth Personal Care School Stuff Dating All Categories Arts and Entertainment Finance and Business Home and Garden Relationship Quizzes Cars & Other Vehicles Food and Entertaining Personal Care and Style Sports and Fitness Computers and Electronics Health Pets and Animals Travel Education & Communication Hobbies and Crafts Philosophy and Religion Work World Family Life Holidays and Traditions Relationships Youth
• Browse Articles
• Learn Something New
• Quizzes Hot
• This Or That Game
• Train Your Brain
• Explore More
• Support wikiHow
• About wikiHow
• Log in / Sign up
• Education and Communications
• Mathematics

How to Solve an Algebraic Expression

Last Updated: April 6, 2024 Fact Checked

This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 10 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 490,858 times.

An algebraic expression is a mathematical phrase that contains numbers and/or variables. Though it cannot be solved because it does not contain an equals sign (=), it can be simplified. You can, however, solve algebraic equations , which contain algebraic expressions separated by an equals sign. If you want to know how to master this mathematical concept, then see Step 1 to get started.

Understanding the Basics

• Algebraic expression : 4x + 2
• Algebraic equation : 4x + 2 = 100

• 3x 2 + 5 + 4x 3 - x 2 + 2x 3 + 9 =
• 3x 2 - x 2 + 4x 3 + 2x 3 + 5 + 9 =
• 2x 2 + 6x 3 + 14

• You can see that each coefficient can be divisible by 3. Just "factor out" the number 3 by dividing each term by 3 to get your simplified equation.
• 3x/3 + 15/3 = 9x/3 + 30/3 =
• x + 5 = 3x + 10

• (3 + 5) 2 x 10 + 4
• First, follow P, the operation in the parentheses:
• = (8) 2 x 10 + 4
• Then, follow E, the operation of the exponent:
• = 64 x 10 + 4
• Next, do multiplication:
• And last, do addition:

• 5x + 15 = 65 =
• 5x/5 + 15/5 = 65/5 =
• x + 3 = 13 =

Joseph Meyer

To solve an equation for a variable like "x," you need to manipulate the equation to isolate x. Use techniques like the distributive property, combining like terms, factoring, adding or subtracting the same number, and multiplying or dividing by the same non-zero number to isolate "x" and find the answer.

Solve an Algebraic Equation

• 4x + 16 = 25 -3x =
• 4x = 25 -16 - 3x
• 4x + 3x = 25 -16 =
• 7x/7 = 9/7 =

• First, subtract 12 from both sides.
• 2x 2 + 12 -12 = 44 -12 =
• Next, divide both sides by 2.
• 2x 2 /2 = 32/2 =
• Solve by taking the square root of both sides, since that will turn x 2 into x.
• √x 2 = √16 =
• State both answers:x = 4, -4

• First, cross multiply to get rid of the fraction. You have to multiply the numerator of one fraction by the denominator of the other.
• (x + 3) x 3 = 2 x 6 =
• Now, combine like terms. Combine the constant terms, 9 and 12, by subtracting 9 from both sides.
• 3x + 9 - 9 = 12 - 9 =
• Isolate the variable, x, by dividing both sides by 3 and you've got your answer.
• 3x/3 = 3/3 =

• First, move everything that isn't under the radical sign to the other side of the equation:
• √(2x+9) = 5
• Then, square both sides to remove the radical:
• (√(2x+9)) 2 = 5 2 =
• Now, solve the equation as you normally would by combining the constants and isolating the variable:
• 2x = 25 - 9 =

• |4x +2| - 6 = 8 =
• |4x +2| = 8 + 6 =
• |4x +2| = 14 =
• 4x + 2 = 14 =
• Now, solve again by flipping the sign of the term on the other side of the equation after you've isolated the absolute value:
• 4x + 2 = -14
• 4x = -14 -2
• 4x/4 = -16/4 =
• Now, just state both answers: x = -4, 3

Community Q&A

• The degree of a polynomial is the highest power within the terms. Thanks Helpful 9 Not Helpful 1
• Once you're done, replace the variable with the answer, and solve the sum to see if it makes sense. If it does, then, congratulations! You just solved an algebraic equation! Thanks Helpful 7 Not Helpful 3
• To cross-check your answer, visit wolfram-alpha.com. They give the answer and often the two steps. Thanks Helpful 8 Not Helpful 5

You Might Also Like

• ↑ https://www.math4texas.org/Page/527
• ↑ https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-expressions-and-variables/cc-6th-combining-like-terms/v/combining-like-terms-2
• ↑ https://www.mathsisfun.com/algebra/factoring.html
• ↑ https://www.mathsisfun.com/operation-order-pemdas.html
• ↑ https://sciencing.com/tips-for-solving-algebraic-equations-13712207.html
• ↑ https://www.mathsisfun.com/algebra/equations-solving.html
• ↑ https://tutorial.math.lamar.edu/Classes/Alg/SolveExpEqns.aspx
• ↑ https://www.mathsisfun.com/algebra/fractions-algebra.html
• ↑ https://math.libretexts.org/Courses/Coastline_College/Math_C045%3A_Beginning_and_Intermediate_Algebra_(Chau_Duc_Tran)/10%3A_Roots_and_Radicals/10.07%3A_Solve_Radical_Equations
• ↑ https://www.mathplanet.com/education/algebra-1/linear-inequalitites/solving-absolute-value-equations-and-inequalities

About This Article

If you want to solve an algebraic expression, first understand that expressions, unlike equations, are mathematical phrase that can contain numbers and/or variables but cannot be solved. For example, 4x + 2 is an expression. To reduce the expression, combine like terms, for example everything with the same variable. After you've done that, factor numbers by finding the lowest common denominator. Then, use the order of operations, which is known by the acronym PEMDAS, to reduce or solve the problem. To learn how to solve algebraic equations, keep scrolling! Did this summary help you? Yes No

• Send fan mail to authors

Reader Success Stories

Marie Taufa

May 27, 2018

Did this article help you?

Sep 16, 2023

Vickram Bheemsain

May 5, 2016

Imran Ahmed

Nov 1, 2018

Veer Gaudani

Feb 21, 2017

Featured Articles

Trending Articles

Watch Articles

• Terms of Use
• Privacy Policy
• Do Not Sell or Share My Info
• Not Selling Info

Don’t miss out! Sign up for

wikiHow’s newsletter

• Solve equations and inequalities
• Simplify expressions
• Factor polynomials
• Graph equations and inequalities
• Advanced solvers
• All solvers
• Arithmetics
• Determinant
• Percentages
• Scientific Notation
• Inequalities

What can QuickMath do?

QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
• The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
• The calculus section will carry out differentiation as well as definite and indefinite integration.
• The matrices section contains commands for the arithmetic manipulation of matrices.
• The graphs section contains commands for plotting equations and inequalities.
• The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

Math Topics

More solvers.

• Add Fractions
• Simplify Fractions

Solving Equations

What is an equation.

An equation says that two things are equal. It will have an equals sign "=" like this:

That equations says:

what is on the left (x − 2)  equals  what is on the right (4)

So an equation is like a statement " this equals that "

What is a Solution?

A Solution is a value we can put in place of a variable (such as x ) that makes the equation true .

Example: x − 2 = 4

When we put 6 in place of x we get:

which is true

So x = 6 is a solution.

How about other values for x ?

• For x=5 we get "5−2=4" which is not true , so x=5 is not a solution .
• For x=9 we get "9−2=4" which is not true , so x=9 is not a solution .

In this case x = 6 is the only solution.

You might like to practice solving some animated equations .

More Than One Solution

There can be more than one solution.

Example: (x−3)(x−2) = 0

When x is 3 we get:

(3−3)(3−2) = 0 × 1 = 0

And when x is 2 we get:

(2−3)(2−2) = (−1) × 0 = 0

which is also true

So the solutions are:

x = 3 , or x = 2

When we gather all solutions together it is called a Solution Set

The above solution set is: {2, 3}

Solutions Everywhere!

Some equations are true for all allowed values and are then called Identities

Example: sin(−θ) = −sin(θ) is one of the Trigonometric Identities

Let's try θ = 30°:

sin(−30°) = −0.5 and

−sin(30°) = −0.5

So it is true for θ = 30°

Let's try θ = 90°:

sin(−90°) = −1 and

−sin(90°) = −1

So it is also true for θ = 90°

Is it true for all values of θ ? Try some values for yourself!

How to Solve an Equation

There is no "one perfect way" to solve all equations.

A Useful Goal

But we often get success when our goal is to end up with:

x = something

In other words, we want to move everything except "x" (or whatever name the variable has) over to the right hand side.

Example: Solve 3x−6 = 9

Now we have x = something ,

and a short calculation reveals that x = 5

Like a Puzzle

In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things we can (and cannot) do.

Here are some things we can do:

• Add or Subtract the same value from both sides
• Clear out any fractions by Multiplying every term by the bottom parts
• Divide every term by the same nonzero value
• Combine Like Terms
• Expanding (the opposite of factoring) may also help
• Recognizing a pattern, such as the difference of squares
• Sometimes we can apply a function to both sides (e.g. square both sides)

Example: Solve √(x/2) = 3

And the more "tricks" and techniques you learn the better you will get.

Special Equations

There are special ways of solving some types of equations. Learn how to ...

• solve Quadratic Equations
• solve Radical Equations
• solve Equations with Sine, Cosine and Tangent

Check Your Solutions

You should always check that your "solution" really is a solution.

How To Check

Take the solution(s) and put them in the original equation to see if they really work.

Example: solve for x:

2x x − 3 + 3 = 6 x − 3     (x≠3)

We have said x≠3 to avoid a division by zero.

Let's multiply through by (x − 3) :

2x + 3(x−3) = 6

Bring the 6 to the left:

2x + 3(x−3) − 6 = 0

Expand and solve:

2x + 3x − 9 − 6 = 0

5x − 15 = 0

5(x − 3) = 0

Which can be solved by having x=3

Let us check x=3 using the original question:

2 × 3 3 − 3 + 3  =   6 3 − 3

Hang On: 3 − 3 = 0 That means dividing by Zero!

And anyway, we said at the top that x≠3 , so ...

x = 3 does not actually work, and so:

There is No Solution!

That was interesting ... we thought we had found a solution, but when we looked back at the question we found it wasn't allowed!

This gives us a moral lesson:

"Solving" only gives us possible solutions, they need to be checked!

• Note down where an expression is not defined (due to a division by zero, the square root of a negative number, or some other reason)
• Show all the steps , so it can be checked later (by you or someone else)

Imaginary Numbers Explained with Formula: $$ \sqrt{-1} \text{= ?} $$

What they are and how to simplify.

$$ i^5 = ? \\ i ^ {21} = ? $$

$$ \sqrt{-25} = ? \\ \sqrt{-18} = ? $$

Video Tutorial on Simplifying Imaginary Numbers

What is an imaginary number anyway?

Imaginary numbers are based on the mathematical number $$ i $$ .

$$ i \text { is defined to be } \sqrt{-1} $$

From this 1 fact, we can derive a general formula for powers of $$ i $$ by looking at some examples.

$ \text{ Table 1} \\ \begin{array}{ccc|c} \hline Expression & & Work & Result \\\hline \red{i^ \textbf{2}} & = & i \cdot i = \sqrt{-1} \cdot \sqrt{-1} & \red{ \textbf{ -1 }} \\\hline \red{i^ \textbf{3}} & = & i^2 \cdot i = -1 \cdot i & \red{ \textbf{-i} } \\\hline \red{ i^ \textbf{4} } & = & i^2 \cdot i^2 -1 \cdot -1 = & \red{1} \\\hline \end{array} $

You should understand Table 1 above .

Table 1 above boils down to the 4 conversions that you can see in Table 2 below. You should memorize Table 2 below because once you start actually solving problems , you'll see you use table 2 over and over again!

What is the larger pattern?

In order to understand how to simplify the powers of $$ i $$, let's look at some more examples, and we'll soon see a formula emerge!

$ \begin{array}{c|c|c} Expression & Work & Result \\\hline \red{i^ \textbf{5}} & \blue{i^4} \cdot i^1 = \blue{1} \cdot i & \red{ \textbf{ i }} \\\hline \red{i^ \textbf{6}} & \blue{i^4} \cdot i^2= \blue{1} \cdot -1 & \red{ \textbf{-1}} \\\hline \red{ i^ \textbf{7} } & \blue{ i^4} \cdot i^3 =\blue{1} \cdot -i & \red{ \boldsymbol{ -i}} \\\hline \red{ i^ \textbf{8} } & = \blue{ i^4} \cdot \blue{ i^4}= \blue{1} \cdot \blue{1} = & \red{ \textbf{ 1}} \\\hline \end{array} $

Do you see the pattern yet? Let's look at 4 more and then summarize.

The General Formula

$$ i^k$$ is the same as $$ i^\red{r} $$ where $$ \red{r} $$ is the remainder when k is divided by 4.

Whether the remainder is 1 , 2 , 3 , or 4 , the key to simplifying powers of i is the remainder when the exponent is divided by 4 .

Practice Problems

$$ i^ {23} $$

Calculate the remainder

$$ 23 \div 4 $$ has a remainder of $$ \red{3} $$

Simplify using Table2

$$ i^{\red{3}} = -i $$

$$ i^ {18} $$

$$ 18 \div 4 $$ has a remainder of $$ \red{2} $$

$$ i^{ \red{2}} $$

$$ i^{ \red{2}} = -1 $$

$$ i^ {41} $$

$$41 \div 4 $$ has a remainder of $$ \red{1} $$

$$ i^{ \red{1}} $$

$$ i^{ \red{1}} = i $$

$$ i^ {100} $$

$$ 100 \div 4 $$ has a remainder of $$ \red{0} $$

$$ i^{ \red{0}} $$

$$ i^{ \red{0}} = 1 $$

$$ 5i^ {22} $$

Remember your order of operations . Exponents must be evaluated before multiplication so you can think of this problem as $$ 5 \cdot (\color{Blue}{i^ {22}}) $$

$$ 22 \div 4 $$ has a remainder of $$ \red{2} $$

Now, the rest

$$ 5 \cdot ( {\color{Blue}-1} ) = -5 $$

$$ 12i^ {36} $$

Remember your order of operations . Exponents must be evaluated before multiplication so you can think of this problem as $$ 12 \cdot ( {\color{Blue}i^ {36}}) $$

$$ 36 \div 4 $$ has a remainder of $$ \red{0} $$

$$ 12 \cdot ( {\color{Blue} 1} ) = 12 $$

$$ 7 i^ {103} $$

Remember your order of operations . Exponents must be evaluated before multiplication so you can think of this problem as $$ 7 \cdot ( {\color{Blue}i^ {103}}) $$

$$ 103 \div 4 $$ has a remainder of $$ \red{3} $$

$$ i^{ \red{3}} $$

$$ i^{ \red{3}} = -i $$

$$ 7 \cdot ( {\color{Blue} -i} ) = -7i $$

• Complex Numbers Home
• Simplify Negative Radicals
• What is a complex Number
• Complex Numbers Worksheets

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

Popular pages @ mathwarehouse.com.

• Math Lessons
• Math Formulas
• Calculators

Math Calculators, Lessons and Formulas

It is time to solve your math problem

• HW Help (paid service)
• Solving Equations
• Step-by-step Equation Solver

Step by step equation solver

This is an online calculator for solving algebraic equations. Simply enter the equation, and the calculator will walk you through the steps necessary to simplify and solve it. Each step is followed by a brief explanation.

• Factoring Polynomials
• Solving equations
• Rationalize Denominator
• Arithmetic sequences
• Polynomial Roots
• Synthetic Division
• Polynomial Operations
• Graphing Polynomials
• Simplify Polynomials
• Generate From Roots
• Simplify Expression
• Multiplication / Division
• Addition / Subtraction
• Simplifying
• Quadratic Equations Solver
• Polynomial Equations
• Solving Equations - With Steps
• Solving (with steps)
• Quadratic Plotter
• Factoring Trinomials
• Equilateral Triangle
• Right Triangle
• Oblique Triangle
• Square Calculator
• Rectangle Calculator
• Circle Calculator
• Hexagon Calculator
• Rhombus Calculator
• Trapezoid Calculator
• Triangular Prism
• Distance calculator
• Midpoint Calculator
• Triangle Calculator
• Graphing Lines
• Lines Intersection
• Two Point Form
• Line-Point Distance
• Parallel/Perpendicular
• Circle Equation
• Circle From 3 Points
• Circle-line Intersection
• Modulus, inverse, polar form
• Vectors (2D & 3D)
• Add, Subtract, Multiply
• Determinant Calculator
• Matrix Inverse
• Characteristic Polynomial
• Eigenvalues
• Eigenvectors
• Matrix Decomposition
• Limit Calculator
• Derivative Calculator
• Integral Calculator
• Arithmetic Sequences
• Geometric Sequences
• Find n th Term
• Degrees to Radians
• Trig. Equations
• Long Division
• Evaluate Expressions
• Fraction Calculator
• Greatest Common Divisor GCD
• Least Common Multiple LCM
• Prime Factorization
• Scientific Notation
• Percentage Calculator
• Dec / Bin / Hex
• Probability Calculator
• Probability Distributions
• Descriptive Statistics
• Standard Deviation
• Z - score Calculator
• Normal Distribution
• T-Test Calculator
• Correlation & Regression
• Simple Interest
• Compound Interest
• Amortization Calculator
• Annuity Calculator
• Work Problems

Hire MATHPORTAL experts to do math homework for you.

Prices start at $3 per problem.

Related Calculators

Was this calculator helpful?

Please tell me how can I make this better.

Welcome to MathPortal. This website's owner is mathematician Miloš Petrović. I designed this website and wrote all the calculators, lessons, and formulas .

If you want to contact me, probably have some questions, write me using the contact form or email me on [email protected]

Email (optional)

Game Central

Inequalities, absolute value and rounding, related concepts.

Equation Solver

What do you want to calculate.

• Solve for Variable
• Practice Mode
• Step-By-Step

Solver Title

Generating PDF...

• Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Expanded Form Mean, Median & Mode
• Algebra Equations Inequalities System of Equations System of Inequalities Basic Operations Algebraic Properties Partial Fractions Polynomials Rational Expressions Sequences Power Sums Interval Notation Pi (Product) Notation Induction Logical Sets Word Problems
• Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
• Calculus Derivatives Derivative Applications Limits Integrals Integral Applications Integral Approximation Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Fourier Transform
• Functions Line Equations Functions Arithmetic & Comp. Conic Sections Transformation
• Linear Algebra Matrices Vectors
• Trigonometry Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify
• Statistics Mean Geometric Mean Quadratic Mean Average Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal Distribution
• Physics Mechanics
• Chemistry Chemical Reactions Chemical Properties
• Finance Simple Interest Compound Interest Present Value Future Value
• Economics Point of Diminishing Return
• Conversions Roman Numerals Radical to Exponent Exponent to Radical To Fraction To Decimal To Mixed Number To Improper Fraction Radians to Degrees Degrees to Radians Hexadecimal Scientific Notation Distance Weight Time Volume
• Pre Algebra
• One-Step Addition
• One-Step Subtraction
• One-Step Multiplication
• One-Step Division
• One-Step Decimals
• Two-Step Integers
• Two-Step Add/Subtract
• Two-Step Multiply/Divide
• Two-Step Fractions
• Two-Step Decimals
• Multi-Step Integers
• Multi-Step with Parentheses
• Multi-Step Rational
• Multi-Step Fractions
• Multi-Step Decimals
• Solve by Factoring
• Completing the Square
• Quadratic Formula
• Biquadratic
• Logarithmic
• Exponential
• Rational Roots
• Floor/Ceiling
• Equation Given Roots
• Newton Raphson
• Substitution
• Elimination
• Cramer's Rule
• Gaussian Elimination
• System of Inequalities
• Perfect Squares
• Difference of Squares
• Difference of Cubes
• Sum of Cubes
• Polynomials
• Distributive Property
• FOIL method
• Perfect Cubes
• Binomial Expansion
• Negative Rule
• Product Rule
• Quotient Rule
• Expand Power Rule
• Fraction Exponent
• Exponent Rules
• Exponential Form
• Logarithmic Form
• Absolute Value
• Rational Number
• Powers of i
• Complex Form
• Partial Fractions
• Is Polynomial
• Leading Coefficient
• Leading Term
• Standard Form
• Complete the Square
• Synthetic Division
• Linear Factors
• Rationalize Denominator
• Rationalize Numerator
• Identify Type
• Convergence
• Interval Notation
• Pi (Product) Notation
• Boolean Algebra
• Truth Table
• Mutual Exclusive
• Cardinality
• Caretesian Product
• Age Problems
• Distance Problems
• Cost Problems
• Investment Problems
• Number Problems
• Percent Problems
• Addition/Subtraction
• Multiplication/Division
• Dice Problems
• Coin Problems
• Card Problems
• Pre Calculus
• Linear Algebra
• Trigonometry
• Conversions

Most Used Actions

Number line.

• x^4-5x^2+4=0
• \sqrt{x-1}-x=-7
• \left|3x+1\right|=4
• \log _2(x+1)=\log _3(27)
• 3^x=9^{x+5}
• What is the completing square method?
• Completing the square method is a technique for find the solutions of a quadratic equation of the form ax^2 + bx + c = 0. This method involves completing the square of the quadratic expression to the form (x + d)^2 = e, where d and e are constants.
• What is the golden rule for solving equations?
• The golden rule for solving equations is to keep both sides of the equation balanced so that they are always equal.
• How do you simplify equations?
• To simplify equations, combine like terms, remove parethesis, use the order of operations.
• How do you solve linear equations?
• To solve a linear equation, get the variable on one side of the equation by using inverse operations.

equation-calculator

• High School Math Solutions – Quadratic Equations Calculator, Part 1 A quadratic equation is a second degree polynomial having the general form ax^2 + bx + c = 0, where a, b, and c...

Please add a message.

Message received. Thanks for the feedback.

Help Center Help Center

• Help Center
• Trial Software
• Product Updates
• Documentation

Solve Algebraic Equations

Symbolic Math Toolbox™ offers both symbolic and numeric equation solvers. This topic shows you how to solve an equation symbolically using the symbolic solver solve . To compare symbolic and numeric solvers, see Select Numeric or Symbolic Solver .

Solve an Equation

Return the full solution to an equation, work with the full solution, parameters, and conditions returned by solve, visualize and plot solutions returned by solve, simplify complicated results and improve performance.

If eqn is an equation, solve(eqn, x) solves eqn for the symbolic variable x .

Use the == operator to specify the familiar quadratic equation and solve it using solve .

solx is a symbolic vector containing the two solutions of the quadratic equation. If the input eqn is an expression and not an equation, solve solves the equation eqn == 0 .

To solve for a variable other than x , specify that variable instead. For example, solve eqn for b .

If you do not specify a variable, solve uses symvar to select the variable to solve for. For example, solve(eqn) solves eqn for x .

solve does not automatically return all solutions of an equation. Solve the equation cos(x) == -sin(x) . The solve function returns one of many solutions.

To return all solutions along with the parameters in the solution and the conditions on the solution, set the ReturnConditions option to true . Solve the same equation for the full solution. Provide three output variables: for the solution to x , for the parameters in the solution, and for the conditions on the solution.

solx contains the solution for x , which is pi*k - pi/4 . The param variable specifies the parameter in the solution, which is k . The cond variable specifies the condition in(k, 'integer') on the solution, which means k must be an integer. Thus, solve returns a periodic solution starting at pi/4 which repeats at intervals of pi*k , where k is an integer.

You can use the solutions, parameters, and conditions returned by solve to find solutions within an interval or under additional conditions.

To find values of x in the interval -2*pi<x<2*pi , solve solx for k within that interval under the condition cond . Assume the condition cond using assume .

To find values of x corresponding to these values of k , use subs to substitute for k in solx .

To convert these symbolic values into numeric values for use in numeric calculations, use vpa .

The previous sections used solve to solve the equation cos(x) == -sin(x) . The solution to this equation can be visualized using plotting functions such as fplot and scatter .

Plot both sides of equation cos(x) == -sin(x) .

Calculate the values of the functions at the values of x , and superimpose the solutions as points using scatter .

( - 0.70710678118654752440084436210485 0.70710678118654752440084436210485 - 0.70710678118654752440084436210485 0.70710678118654752440084436210485 )

As expected, the solutions appear at the intersection of the two plots.

If results look complicated, solve is stuck, or if you want to improve performance, see, Troubleshoot Equation Solutions from solve Function .

Related Topics

• Solve System of Linear Equations
• Solve Differential Equation
• Solve Differential Algebraic Equations (DAEs)

MATLAB Command

You clicked a link that corresponds to this MATLAB command:

Run the command by entering it in the MATLAB Command Window. Web browsers do not support MATLAB commands.

Select a Web Site

Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select: .

• Switzerland (English)
• Switzerland (Deutsch)
• Switzerland (Français)
• 中国 (English)

You can also select a web site from the following list:

How to Get Best Site Performance

Select the China site (in Chinese or English) for best site performance. Other MathWorks country sites are not optimized for visits from your location.

• América Latina (Español)
• Canada (English)
• United States (English)
• Belgium (English)
• Denmark (English)
• Deutschland (Deutsch)
• España (Español)
• Finland (English)
• France (Français)
• Ireland (English)
• Italia (Italiano)
• Luxembourg (English)
• Netherlands (English)
• Norway (English)
• Österreich (Deutsch)
• Portugal (English)
• Sweden (English)
• United Kingdom (English)

Asia Pacific

• Australia (English)
• India (English)
• New Zealand (English)

Contact your local office

• Add to Home
• TigerMilk.Education GmbH
• Privacy policy
• Terms of service

Copyright Ⓒ 2013-2024 tiger-algebra.com

• Arabic | اللغة العربية
• Spanish | Español
• French | Français
• German | Deutsch
• Hebrew | עִברִית
• Italian | Italiano
• Indonesian | Bahasa Indonesia
• Portuguese | Português
• Russian | Русский
• Serbian | Srpski
• Vietnamese | Tiếng Việt
• Chinese | 中文
• Hindi | हिंदी
• Bengali | বাংলা
• Filipino | Filipino
• Korean | 한국인
• Marathi | मराठी
• Tamil | தமிழ்
• Telugu | తెలుగు
• Turkish | Türkçe
• Japanese | 日本語

This site is best viewed with Javascript. If you are unable to turn on Javascript, please click here .

Solution - Linear equations with one unknown

Other Ways to Solve

• Alternative approach

Video Explanation

• Watch a video of the solution to (4x/9)-25=-x/4

Step-by-step explanation

1. group all x terms on the left side of the equation.

4 x 9 - 25 = - x 4

Add 1 4 x to both sides:

( 4 x 9 - 25 ) + 1 4 · x = ( - x 4 ) + 1 4 x

Group like terms:

( 4 x 9 + 1 4 · x ) - 25 = ( - x 4 ) + 1 4 x

Group the coefficients:

( 4 9 + 1 4 ) x - 25 = ( - x 4 ) + 1 4 x

Find the lowest common denominator:

( ( 4 · 4 ) ( 9 · 4 ) + ( 1 · 9 ) ( 4 · 9 ) ) x - 25 = ( - x 4 ) + 1 4 x

Multiply the denominators:

( ( 4 · 4 ) 36 + ( 1 · 9 ) 36 ) x - 25 = ( - x 4 ) + 1 4 x

Multiply the numerators:

( 16 36 + 9 36 ) x - 25 = ( - x 4 ) + 1 4 x

Combine the fractions:

( 16 + 9 ) 36 · x - 25 = ( - x 4 ) + 1 4 x

Combine the numerators:

25 36 · x - 25 = ( - x 4 ) + 1 4 x

25 36 · x - 25 = ( - 1 + 1 ) 4 x

25 36 · x - 25 = 0 4 x

Reduce the zero numerator:

25 36 x - 25 = 0 x

Simplify the arithmetic:

25 36 x - 25 = 0

2. Group all constants on the right side of the equation

Add 25 to both sides:

( 25 36 x - 25 ) + 25 = 0 + 25

25 36 x = 0 + 25

25 36 x = 25

3. Isolate the x

Multiply both sides by inverse fraction 36 25 :

( 25 36 x ) · 36 25 = 25 · 36 25

( 25 36 · 36 25 ) x = 25 · 36 25

Multiply the coefficients:

( 25 · 36 ) ( 36 · 25 ) x = 25 · 36 25

Simplify the fraction:

x = 25 · 36 25

Multiply the fraction(s):

x = ( 25 · 36 ) 25

How did we do?

Why learn this

Linear equations cannot tell you the future, but they can give you a good idea of what to expect so you can plan ahead. How long will it take you to fill your swimming pool? How much money will you earn during summer break? What are the quantities you need for your favorite recipe to make enough for all your friends? Linear equations explain some of the relationships between what we know and what we want to know and can help us solve a wide range of problems we might encounter in our everyday lives.

Terms and topics

• Linear equations with one unknown

Related links

• Linear Equations - Math is fun
• Solving equations &amp; inequalities | Khan Academy
• Linear Equations in the Real World | MyWhyU (youtube)

Latest Related Drills Solved

• − 43 = x 5 − 47
• x 2 − 3 x 4 + 5 x 6 = 21
• x 3 − 2 = 6
• 4 ( y − 3 ) + 19 = 8 ( 2 y + 3 ) + 7
• 4 ( 1 / 2 x - 1 ) = 0
• 76 = 10 x − 44
• − 40 = − 5 c − 25
• − 48 − x = − 39
• 9 d − 3 d + 2 d − 7 d + 2 d = 18
• 50 − a 11 = 58