Let's define variables to set up our system of equations:

• Let $x$ represent the number of minutes each one worked.
• Let $y$ represent the amount each one earned.

Given information:

1. Ian's earnings: Ian charges $67 for the visit and $1 per minute, so his total earnings equation is: $y = 67 + 1 \cdot x$

2. Tommy's earnings: Tommy charges $54 for the visit and $2 per minute, so his total earnings equation is: $y = 54 + 2 \cdot x$

Since Ian and Tommy both earned the same amount and worked the same amount of time, we can set up the following system:

$y = 67 + x$ $y = 54 + 2x$

Solving the System by Substitution

Since both equations are set equal to $y$, we can set the right sides equal to each other:

$67 + x = 54 + 2x$

1. Solve for $x$ by isolating it: $67 - 54 = 2x - x$ $13 = x$

2. Substitute $x = 13$ back into one of the original equations to find $y$: $y = 67 + 13$ $y = 80$

Solution

• Each one worked for 13 minutes.
• Each one earned $80.

Final Answer

Ian and Tommy each earned $80 by working for 13 minutes.

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Would you like more details on the solution steps? Here are five related questions:

1. How would the system of equations change if Ian charged $2 per minute and Tommy charged $1 per minute?
2. What if Tommy’s visit fee were increased to $60 but his per-minute rate stayed the same?
3. How would you solve this problem if they worked different amounts of time?
4. How can you verify that both solutions make the equations true?
5. How would the equations change if Ian’s and Tommy’s initial visit fees were equal?

Tip: When both equations are solved for the same variable, setting them equal is a useful substitution method.