Model comparison problems using systems of linear equations

Detailed Explanation: Model comparison problems using systems of linear equations

A comparison problem gives two rules for the same situation. To model it:

1. Choose variables and state what they mean.
2. Write one linear equation for each option.
3. Solve the system. The solution makes both equations true.
4. Interpret the answer using the units and realistic restrictions, such as $x\ge 0$.

Worked example

A gym offers two membership plans:

• Plan A: a $10 sign-up fee plus$2 per visit
• Plan B: a $22 sign-up fee plus$1 per visit

After how many visits will the plans cost the same?

1. Define the variables

Let

• $x=$ number of visits
• $y=$ total cost in dollars

Since the number of visits cannot be negative, $x\ge 0$.

2. Write an equation for each plan

Plan A has a fixed cost of $10$ and costs $2$ for each visit:

Plan B has a fixed cost of $22$ and costs $1$ for each visit:

Together, these equations form a system:

3. Find when the costs are equal

Both equations equal $y$, so set their right sides equal:

Subtract $x$ from both sides:

Subtract $10$:

The plans cost the same after $12$ visits.

4. Find the equal cost

Substitute $x=12$ into either equation:

The solution is

This means the two plans have the same cost—34$—after$12$ visits.

Graphically, the two lines $y=10+2x$ and $y=22+x$ intersect at $(12,34)$. For fewer than $12$ visits, Plan A costs less because it has the lower sign-up fee. For more than $12$ visits, Plan B costs less because it has the lower cost per visit.