3.5 Use a General Strategy to Solve Linear Equations
3.5 Use a General Strategy to Solve Linear Equations
Learning Objectives
By the end of this section, you will be able to:
- Solve equations using a general strategy
- Classify equations
Solve Equations Using the General Strategy
Until now we have dealt with solving one specific form of a linear equation. It is time now to lay out one overall strategy that can be used to solve any linear equation. Some equations we solve will not require all these steps to solve, but many will.
Beginning by simplifying each side of the equation makes the remaining steps easier.
EXAMPLE 1. How to Solve Linear Equations Using the General Strategy
| Step 1. Simplify each side of the equation as much as possible. | Use the Distributive Property.
Notice that each side of the equation is simplified as much as possible. |
|
| Step 2. Collect all variable terms on one side of the equation. | Nothing to do as all the x’s are on the left side. | |
| Step 3. Collect constant terms on the other side of the equation. | To get constants only on the right side, add 18 to each side.
Simplify. |
|
| Step 4. Make the coefficient of the variable term to equal to 1. | Divide each side by .
Simplify |
|
| Step 5. Check the solution. | Let |
TRY IT 1.1
Solve: .
Show answer
TRY IT 1.2
Solve: .
Show answer
General strategy for solving linear equations.
- Simplify each side of the equation as much as possible.
Use the Distributive Property to remove any parentheses.
Combine like terms. - Collect all the variable terms on one side of the equation.
Use the Addition or Subtraction Property of Equality. - Collect all the constant terms on the other side of the equation.
Use the Addition or Subtraction Property of Equality. - Make the coefficient of the variable term to equal to 1.
Use the Multiplication or Division Property of Equality.
State the solution to the equation. - Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.
EXAMPLE 2
Solve: .
Given equation.
Simplify each side of the equation by distributing.
The only term is on the left side, so all variable terms are on the left side of the equation.
Add to both sides to get all constant terms on the right side of the equation.
Simplify.
Rewrite as .
Make the coefficient of the variable term to equal to by dividing both sides by .
Simplify.
Check:
Let .
TRY IT 2.1
Solve: .
Show answer
TRY IT 2.2
Solve: .
Show answer
EXAMPLE 3
Solve: .
Simplify each side of the equation as much as possible.
Distribute.
Combine like terms.
Add 10 to both sides to get all constant terms on the other side of the equation.
Simplify.
Make the coefficient of the variable term to equal to 1 by dividing both sides by 5.
Simplify.
Check: Let
TRY IT 3.1
Solve: .
Show answer
TRY IT 3.2
Solve: .
Show answer
EXAMPLE 4
Solve: .
Distribute.
Simplify each side of the equation.
Add to get all the variables on the left side and simplify.
Add to get constants only on the right and simplify.
Make the coefficient o