Equations with Variables on Both Sides

Unit Objective

● To solve equations with variables on both sides

Key Vocabulary

● Variable - a symbol (usually a letter) standing in for an unknown numerical value in an equation or an algebraic expression. In simple words, a variable is a quantity that can be changed and is not fixed.

● Solution - any value of the variable that satisfies the equality. (It will be the answer to the equation)

● Constant term - a term without a variable. It is simply just a number.

Solve with Variables on Both Sides
To solve equations with variables on both sides, collect the variable terms on one side and the constant terms on the other.

Example 1: Solve 15 – 2x = -7x. 15 – 2x = -7x Write the equation. + 2x + 2x Addition Property of Equality 15 = -5x Simplify. Division Property of Equality -3 = x Simplify. The solution is x = -3.

Let's practice.

1) -3x = 2x +20

2) 2.5y – 6= 4.5y – 1

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Use the Distributive Property to Solve an Equation

Example 1: Solve -2(x – 5) = 6(2 – 0.5x).

-2(x – 5) = 6(2 – 0.5x) Write the equation.

-2x + 10 = 12 – 3x Distributive Property

+ 3x + 3x Addition Property of Equality

x + 10 = 12 Simplify.

– 10 – 10 Subtraction Property of Equality

      x = 2 Simplify.


The solution is x = 2.

Let's practice.

1) 6(4 – z) = 2z

2) 5(w – 2) = -2(1.5w – 5)

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Solving an Equation with No Solution

Example 1: Solve 3 – 4x = -7 – 4x.

3 – 4x = -7 – 4x Write the equation.

+ 4x + 4x Addition Property of Equality

3 = -7 Simplify.


The equation 3 = -7 is never true. So, the equation has no solution.

Let’s practice.

1) 2x + 1 = 2x – 1

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Solve an Equation with Infinitely Many Solutions

Solve Write the equation. 6x + 4 = 6x + 4 Distributive Property. – 6x – 6x Subtraction Property of Equality 4 = 4  Simplify. The equation 4 = 4 is always true. So, the equation has infinitely many solutions.

Let’s Practice.

1) Solve

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Modeling in Real Life
Example 1: A boat travels x miles per hour upstream on the Mississippi River. On the return trip, the boat travels 2 miles per hour faster. How far does the boat travel upstream?

The boat travels the same distance upstream as on the return trip. The speed of the boat on the return trip is (x + 2) miles per hour. Write and solve an equation to find the distance upstream.

x(3) = (x + 2)(2.5) Write an equation. 3x = 2.5x + 5 Distributive Property – 2.5x –2.5x Subtraction Property of Equality 0.5x = 5 Simplify. Division Property of Equality x = 10 Simplify. The boat travels 10 miles per hour for 3 hours upstream. So, the boat travels miles upstream.