# Solving Linear Equations and Inequalities

## Use a General Strategy to Solve Linear 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.

### Classify Equations

Consider the equation we solved at the start of the last section, . The solution we found was . This means the equation  is true when we replace the variable, x, with the value . We showed this when we checked the solution  and evaluated  for .
If we evaluate  for a different value of x, the left side will not be .
The equation  is true when we replace the variable, x, with the value , but not true when we replace x with any other value. Whether or not the equation  is true depends on the value of the variable. Equations like this are called conditional equations.
All the equations we have solved so far are conditional equations.
Now let’s consider the equation . Do you recognize that the left side and the right side are equivalent? Let’s see what happens when we solve for y.
But  is true.
This means that the equation  is true for any value of y. We say the solution to the equation is all of the real numbers. An equation that is true for any value of the variable like this is called an identity.
What happens when we solve the equation ?
But .
Solving the equation  led to the false statement . The equation  will not be true for any value of z. It has no solution. An equation that has no solution, or that is false for all values of the variable, is called a contradiction.

### Key Concepts

1. General Strategy for Solving Linear Equations



### Practice Makes Perfect

Solve Equations Using the General Strategy for Solving Linear Equations
In the following exercises, solve each linear equation.
Classify Equations
In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?