Linear Equation Solver for
Solving Equations with 1 Variable

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The Linear Equation Solver on this page will solve most one-variable equations containing integers, fractions, and/or decimals.

This free online linear equation calculator will use the distributive and equality properties -- as well as combining like terms and inverse operations -- to solve math linear equations having one variable.

Specifically, this math equation solver will handle multiple distributions (maximum of 2 terms within each parenthesis) and simplifications involving the elimination of fractions and decimals (not radicals or exponents).

Plus, unlike other online math solvers, this calculator will show its work and give a detailed step-by-step explanation as to how it arrived at the result. This makes the solver especially useful for checking the answers you derive from solving the equations manually.

What is a Linear Equation?

A linear equation is an expression that can be written in the form ax+b=0, where a and b are constants and x represents an unknown variable that, once found, will make the expression true. The following are some basic examples of linear equations having one variable:

7x + 3 = 20

2(y + 1) - 4 = 4(y - 3)

16.5a + .25a = 25.1a

  1 x + 3 = 2 x - 4  
2 4 5

How to Solve Linear Equations

Solving linear equations is a process aimed at getting the variable by itself on one side of an equal sign and everything else on the other side. In order to explain how to solve linear equations, I will use an example equation that contains all 4 types of terms that can be handled by the linear equation solver.

Example equation: 3(x-1)+1/2=.25+4
First use the distributive property to get rid of the parenthesis: 3(x-1) = 3x-3
After removing parenthesis we are left with: 3x-3+1/2=.25+4
Remove the fractions by multiplying both sides of the equation by 2 (least common multiple of all denominators in equation): 2(3x-3+1/2)=2(.25+4)
After multiplying both sides by 2 we are left with: 6x-6+1=0.5+8
Remove decimals by multiplying both sides of the equation by 100: 100(6x-6+1)=100(0.5+8)
After multiplying both sides by 100 we are left with: 600x-600+100=50+800
Combine the like terms: -600 + 100 = -500 and 50 + 800 = 850
After combining like terms we are left with: 600x-500=+850
Add 500 to both sides of the above equation to get constants on right side: -500 + 500 = 0 and 850 + 500 = 1350
After adding 500 to both sides we are left with: 600x=1350
Get the variable by itself by dividing both sides by 600:
600x = 1350
600 600
After dividing both sides by 600 we are left with:
x = 1350
600
Simplify the result:
x = 1350 = 1350 ÷ 150 = 9 or 2 1
600 600 ÷ 150 4 4

Always Check to Make Sure Your Answer is a Solution

Regardless of which method or tool you use to solve linear equations, you should always check your results by substituting the result for the variable and seeing if the resulting equation proves to be true. After all, humans create math solvers, and humans make mistakes.

In order to make sure our answer to the above example linear equation is in fact a solution, let's substitute 9/4 for x and check to see if the resulting equation proves to be true:

3(9/4 - 1) + 1/2 = .25 + 4

4(27/4 - 3/1 + 1/2) = 4(4.25)

27 - 12 + 2 = 17

15 + 2 = 17

17 = 17

The solution 9/4 is the solution to the example equation.

With that, let's use the Linear Equation Solver for solving linear equations using the distributive property, the addition, subtraction, multiplication, and division properties of equality, as well the necessary inverse operations.

Linear Equation Solver
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Instructions: Enter a linear equation that fits within the following current guidelines.

  • No spaces between terms or operators.
  • One variable only and no radicals or exponents.
  • Parenthesis only for distribution and no more than 2 terms within a parenthesis, and no nested parenthesis (parenthesis inside parenthesis).
  • Numerators and denominators must be integers only (no expressions).
  • Decimal numbers can have a maximum of 2 decimal points.

Next, click the "Solve Linear Equation" button.

IMPORTANT! Since it is impossible to test all possible equations with all web browsers and version releases, be sure to substitute the calculated result for the variable and solve the resulting equation to make sure the calculated result is in fact a solution for the equation.

Mouse over the blue question marks for a further explanation of each entry field or button. More in-depth explanations can be found in the glossary of terms located beneath the Linear Equation Solver.

Help Example linear equations:
Help Linear equation:
Help Calculated solution:

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Linear Equation Solver Calculator Glossary of Terms

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Distributive Property: The expression a(b + x) is the same as ab + ax. Therefore, 5(2 + 5) simplifies to 10 + 25 (5 * 2 + 5 * 5).

Addition and Subtraction Properties of Equality: If the expressions on each side of an equation are equal, then adding or subtracting the same amount from both sides of the equation will not change the equation.

Multiplication and Division Properties of Equality: If the expressions on each side of an equation are equal, then multiplying or dividing by the same amount (except for zero) on both sides of the equation will not change the equation.

Examples of linear Equations: To see an example of how the linear equation solver calculates a solution, as well as what types of equations it accepts, select an example equation from the drop down menu - which will enter the selection in the equation field for solving.

Linear equation: Enter an equation that fits within the guidelines stated at the top of the linear equation solver. Note that distributions may contain only 2 terms within the parenthesis [a(x+1) not a(x+1+4)].

Calculated solution: This is the result calculated by the linear equation solver for the entered equation. Since it is impossible to test the linear equation solver with all possible equations, please be sure to substitute the calculated result for the variable to make sure the result is in fact a solution to the equation.

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