# Solving Linear Equations and Inequalities

## Solve Equations with Variables and Constants on Both Sides


### Solve Equations with Constants on Both Sides

In all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we will learn to solve equations in which the variable terms, or constant terms, or both are on both sides of the equation.
Our strategy will involve choosing one side of the equation to be the “variable side”, and the other side of the equation to be the “constant side.” Then, we will use the Subtraction and Addition Properties of Equality to get all the variable terms together on one side of the equation and the constant terms together on the other side.
By doing this, we will transform the equation that began with variables and constants on both sides into the form  We already know how to solve equations of this form by using the Division or Multiplication Properties of Equality.

### Solve Equations with Variables on Both Sides

What if there are variables on both sides of the equation? For equations like this, begin as we did above—choose a “variable” side and a “constant” side, and then use the subtraction and addition properties of equality to collect all variables on one side and all constants on the other side.

### Solve Equations with Variables and Constants on Both Sides

The next example will be the first to have variables and constants on both sides of the equation. It may take several steps to solve this equation, so we need a clear and organized strategy.
We’ll list the steps below so you can easily refer to them. But we’ll call this the ‘Beginning Strategy’ because we’ll be adding some steps later in this chapter.
In Step 1, a helpful approach is to make the “variable” side the side that has the variable with the larger coefficient. This usually makes the arithmetic easier.
In the last example, we could have made the left side the “variable” side, but it would have led to a negative coefficient on the variable term. (Try it!) While we could work with the negative, there is less chance of errors when working with positives. The strategy outlined above helps avoid the negatives!
To solve an equation with fractions, we just follow the steps of our strategy to get the solution!
We will use the same strategy to find the solution for an equation with decimals.

### Key Concepts

1. Beginning Strategy for Solving an Equation with Variables and Constants on Both Sides of the Equation



### Practice Makes Perfect

Solve Equations with Constants on Both Sides
In the following exercises, solve the following equations with constants on both sides.
Solve Equations with Variables on Both Sides
In the following exercises, solve the following equations with variables on both sides.
Solve Equations with Variables and Constants on Both Sides
In the following exercises, solve the following equations with variables and constants on both sides.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?