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# Solving Linear Equations

Definitions

An equation is formed when two expressions are linked by an equals sign.

For example:

7(x - 2) = 3x - 9 + 5x

To solve an equation means to find the values of the variable that make the equation a true statement. Those values are called the solution of the equation.

For example, x = -5 is the solution of 7(x - 2) = 3x - 9 + 5x . When we replace x with -5 and simplify, the result is a true statement.

 Original equation. 7(x - 2) = 3x - 9 + 5x Replace x with -5. 7(-5 - 2) = 3(-5) - 9 + 5(-5) Simplify. 7(-7) = - 15 - 9 - 25 The result is a true statement. - 49 = - 49

A linear equation in one variable is an equation that can be written in the form ax + b = 0 where a and b are real numbers, and a 0.

Note:

The equation ax + b = 0 is a linear equation because the exponent of the variable, x, is 1.

## Solving a Linear Equation

To find the solution of a linear equation, isolate the variable. That is, use algebraic operations to get the variable by itself on one side of the equals sign.

Here is the procedure to solve a linear equation.

Procedure

To Solve a Linear Equation

Step 1 Remove any parentheses using the Distributive Property.

Step 2 On each side of the equation, combine like terms.

Step 3 Isolate the variable.

Step 4 Check the solution in the original equation.

Note:

The distributive property states that for all real numbers, a, b, c: a(b + c) = ab + ac

Example 1

Solve: 3x + 7 - 8x = -13

Solution

 Step 1 Remove parentheses. There are no parentheses to remove. Step 2 On each side of the equation, combine like terms. Combine 3x and -8x. -5x + 7 - 8x -5x + 7 = -13 = -13 Step 3 Isolate the variable. Subtract 7 from both sides. Simplify. Divide both sides by -5. -5x + 7 - 7-5x x = -13 - 7= -20 = 4 Step 4 Check the solution in the original equation. x = 4 is a solution if it makes the equation true.
 IsIs Is 3x + 7 3(4) + 7 12 + 7 - 8x- 8(4) - 32 - 13 = -13= -13 ? = -13 ? = -13 ? Yes

The solution is x = 4.