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. 


Is Is
Is 
3x + 7
3(4) + 7
12 + 7

 8x  8(4)
 32
 13 
= 13 = 13 ?
= 13 ?
= 13 ? Yes 
The solution is x = 4.
