2. Solving Linear Equations and Inequalities
2.2 Use a General Strategy to Solve Linear Equations
Learning Objectives
By the end of this section it is expected that 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
Solve:
TRY IT 1
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:
| Simplify each side of the equation as much as possible 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
Solve:
Show answer
EXAMPLE 3
Solve: .
| Simplify each side of the equation as much as possible. | |
| Distribute. | |
| Combine like terms. | |
| The only term is on the left side, so all variable terms are on one side of the equation. | |
| Add 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 by dividing both sides by . | |
| Simplify. | |
| Check: | |
| Let . | |
TRY IT 3
Solve: .
Show answer
EXAMPLE 4
Solve: .
| Distribute. | |
| Add to get the variables only to the left. | |
| Simplify. | |
| Add to get constants only on the right. | |
| Simplify. | |
| Divide by . | |
| Simplify. | |
| Check: | |
| Let . | |
TRY IT 4
Solve: .
Show answer
EXAMPLE 5
Solve: .
| Simplify—use the Distributive Property. | |
| Combine like terms. | |
| Add to both sides to collect constants on the right. | |
| Simplify. | |
| Divide both sides by . | |
| Simplify. | |
| Check: Let |
TRY IT 5
Solve: .
Show answer
EXAMPLE 6
Solve: .
| Distribute. | |
| Combine like terms. | |
| Subtract to get the variables only on the right side since > . | |
| Simplify. | |
| Subtract to get the c