Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Frank is trying to incorporate more exercise into his busy schedule. He has several short exercise routines he can complete at home. Last week, he worked out for a total of 138 minutes by doing 3 arm routines and 3 abdominal routines. This week, he has completed 2 arm routines and 3 abdominal routines and spent a total of 119 minutes exercising. How long does each routine last?

An arm routine takes how many minutes to complete, and an abdominal routine takes how many
minutes to complete.

Answer 1

Let's define the following variables:

- Let \( x \) be the number of minutes an arm routine takes to complete.
- Let \( y \) be the number of minutes an abdominal routine takes to complete.

We can model the information given in the problem as a system of equations. We know:

1. Last week, Frank completed 3 arm routines and 3 abdominal routines for a total of 138 minutes:
\[
3x + 3y = 138
\]
2. This week, he completed 2 arm routines and 3 abdominal routines for a total of 119 minutes:
\[
2x + 3y = 119
\]

We can simplify the first equation by dividing everything by 3:
\[
x + y = 46 \tag{1}
\]

Now we have the system of equations:
\[
\begin{align*}
x + y &= 46 \tag{1} \\
2x + 3y &= 119 \tag{2}
\end{align*}
\]

We can solve this system using the substitution or elimination method. Let's use substitution:

From equation (1), we can express \( y \) in terms of \( x \):
\[
y = 46 - x
\]

Now we will substitute this expression for \( y \) into equation (2):
\[
2x + 3(46 - x) = 119
\]

Simplifying this gives:
\[
2x + 138 - 3x = 119
\]
\[
-1x + 138 = 119
\]
\[
-x = 119 - 138
\]
\[
-x = -19
\]
\[
x = 19
\]

Now, substituting \( x = 19 \) back into equation (1) to find \( y \):
\[
y = 46 - 19
\]
\[
y = 27
\]

Thus, the solutions are:
- The arm routine takes \( x = 19 \) minutes.
- The abdominal routine takes \( y = 27 \) minutes.

**Answer:**

An arm routine takes **19 minutes** to complete, and an abdominal routine takes **27 minutes** to complete.