Solving Linear Equations
Strategy for Solving Linear Equations
The most basic equations of algebra are linear equations. There is even a
connection between linear equations in one variable and straight lines.
Linear Equation in One Variable
A linear equation in one variable x is an equation of
the form ax + b = 0, where a and b are real numbers, with a ≠ 0.
A linear equation has exactly one solution. The strategy that
we use for solving linear equations is summarized in the following lines.
Strategy for Solving a Linear Equation
1. If fractions are present, multiply each side by the LCD to
eliminate them.
2. Use the distributive property to remove parentheses.
3. Combine any like terms.
4. Use the addition property of equality to get all variables
on one side and numbers on the other side.
5. Use the multiplication property of equality to get a
single variable on one side.
6. Check by replacing the variable in the original equation
with your solution.
Note that not all equations require all of the steps.
Example 1
Using the equationsolving strategy
Solve the equation
Solution
We first multiply each side of the equation by 10, the LCD
for 2, 5, and 10. However, we do not have to write down that step. We can simply
use the distributive property to multiply each term of the equation by 10.





Multiply each side by 10. 
5y  2(y  4) 
= 23 
Divide each denominator into 10 to
eliminate fractions. 
5y  2y + 8 
= 23 
Be careful to change all signs: 2(y  4)
= 2y + 8 
3y + 8 
= 23 
Combine like terms. 
3y + 8  8 
= 23  8 
Subtract 8 from each side. 
3y 
= 15 
Simplify 


Divide each side by 3. 
y 
= 5 

Check that 5 satisfies the original equation. The solution
set is {5}.
