Equation Word Problems with Variables on Both Sides

In the real world, we often need to compare two different options to see when they are exactly the same. For example, you might want to know when two different cell phone plans cost the same amount, or when two people saving money at different rates will have the exact same bank balance.

Because both options are changing over time (or per item), setting them equal to each other creates an equation with a variable on both sides of the equals sign.

How to Set Up the Equation

To solve these problems, follow a simple pattern for both options:

1. Identify the initial value (the starting amount or flat fee).
2. Identify the rate of change (the amount that changes per week, per mile, per item, etc.). This will be multiplied by your variable.
3. Write an expression for both options: $\text{Initial Value} + (\text{Rate} \cdot x)$.
4. Set the two expressions equal to each other and solve for $x$.

Example 1: Comparing Costs

The Problem: Taxi Company A charges a $5$ dollars base fee plus $2$ dollars per mile. Company B charges a $3$ dollars base fee plus $3$ dollars per mile. After how many miles is the cost the same?

The Setup: Let $x$ represent the number of miles.

• Company A's cost: $5 + 2x$
• Company B's cost: $3 + 3x$

Set them equal to find when the costs are the same: $5 + 2x = 3 + 3x$

The Solution: Subtract $2x$ from both sides to get the variables on one side: $5 = 3 + x$

Subtract $3$ from both sides to isolate $x$: $2 = x$

The cost will be exactly the same after $2$ miles.

Example 2: Comparing Savings

The Problem: Jake has $120$ dollars and saves $15$ dollars per week. Maria has $45$ dollars and saves $25$ dollars per week. When will they have the same amount of money?

The Setup: Let $w$ represent the number of weeks.

• Jake's money: $120 + 15w$
• Maria's money: $45 + 25w$

Set the two expressions equal: $120 + 15w = 45 + 25w$

The Solution: Subtract $15w$ from both sides: $120 = 45 + 10w$

Subtract $45$ from both sides: $75 = 10w$

Divide by $10$: $7.5 = w$

Jake and Maria will have the exact same amount of money in $7.5$ weeks.