# 2.4 Solving Equations with Variables on Both Sides

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2.4 Solving Equations with Variables on Both Sides
I can solve equations with variables on both sides and identify equations that are identities or have no solution.

How to Solve Use properties of equality and inverse operations to produce simpler equivalent equations. Get variables on one side (choose one) Isolate the variable Use inverse operations Check

Example 5x + 2 = 2x +14 5x + 2 – 2x = 2x + 14 - 2x 3x +2 = 14
3𝑥 3 = 12 3 x = 4

Using an Equation with Variables on Both sides
It takes a graphic designer 1.5 h to make one page of a website. With new software, he could complete a page in 1.25 h, but it takes 8 h to learn. How many pages would he have to make in order to save time using the new software? What do we know? What are we looking for? Make an equation 1.5p = 1.25p + 8 p= 32 means with 32 pages the time is the same

Solving Using the Distributive Property
Distribute first, then solve. Examples: 2(5x – 1) = 3(x + 11) 4(2y + 1) = 2(y – 13)

Identities An equation that is true for every possible value of the variable is an identity. This can happen when variables cancel and you are left with a true statement. Ex: x + 1 = x + 1 After moving variables to one side, you are left with 1 = 1, which is true.

Equations with No Solution
When variables cancel and you are left with a false statement, the equation has no solution. Ex: x + 2 = x – 4 After moving variables to one side, you are left with 2 = -4, which is false.

Practice Tell whether the equation is an identity or has no solution.
10x +12 = 2(5x + 6) 9m – 4 = -3m m 3y – 5 = 3(-2 + y)

Assignment ODDS ONLY P.105 #11-45