2.4 - Solve Equations with Variables on Both Sides

• To solve equations with variables on both sides

• To identify equations that are identities or have no solution

Our goal in solving an equation is to isolate the variable (we want it to look like x = some number). We have two tools we can use:

• Inverse operations ("undo")

• Properties of equality ("both sides")

But what if there are variables on both sides of the equation?

Shirley is going to have the exterior of her home painted. Tim’s painting charges $250 plus $14 per hour. Colorful Paints charges $122 per hour. How many hours would the job need to take for Tim’s Painting to be the better deal?

​Colorful Paints

​Tim's Painting

​So Tim's Painting is a better deal for a job that takes three or more hours.

​12 = 12 is obviously true, but there's no variable. This means the equation is true no matter what value we use for x. This kind of equation is called an identity.

x = a number (One solution)

a number equals itself (Identity)

a number equals some other number (No solution)