Question
Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.
William 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 110 minutes by doing 2 arm routines and 3 abdominal routines. This week, he has completed 4 arm routines and 4 abdominal routines and spent a total of 160 minutes exercising. How long does each routine last?
An arm routine takes
minutes to complete, and an abdominal routine takes
minutes to complete.
William 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 110 minutes by doing 2 arm routines and 3 abdominal routines. This week, he has completed 4 arm routines and 4 abdominal routines and spent a total of 160 minutes exercising. How long does each routine last?
An arm routine takes
minutes to complete, and an abdominal routine takes
minutes to complete.
Answers
GPT-4o mini
Let's define the variables:
- Let \( a \) be the time (in minutes) for one arm routine.
- Let \( b \) be the time (in minutes) for one abdominal routine.
From the problem, we can create the following system of equations based on the information provided:
1. From last week:
\[ 2a + 3b = 110 \] (Equation 1)
2. From this week:
\[ 4a + 4b = 160 \] (Equation 2)
Now we can simplify Equation 2. Dividing the entire equation by 4 gives:
\[ a + b = 40 \] (Equation 3)
Now we have the following system of equations:
1. \( 2a + 3b = 110 \)
2. \( a + b = 40 \)
We can solve this system using the substitution or elimination method. Here, we'll use substitution:
From Equation 3, we can express \( b \) in terms of \( a \):
\[ b = 40 - a \]
Now, we substitute \( b \) into Equation 1:
\[ 2a + 3(40 - a) = 110 \]
Expanding the equation:
\[ 2a + 120 - 3a = 110 \]
Now, combine like terms:
\[ -a + 120 = 110 \]
Subtracting 120 from both sides gives:
\[ -a = -10 \]
Multiplying by -1, we get:
\[ a = 10 \]
Now that we have \( a \), we can substitute it back into Equation 3 to find \( b \):
\[ b = 40 - a = 40 - 10 = 30 \]
So the solutions to our variables are:
- An arm routine takes \( a = 10 \) minutes to complete.
- An abdominal routine takes \( b = 30 \) minutes to complete.
Now we fill in the blanks:
An arm routine takes **10** minutes to complete, and an abdominal routine takes **30** minutes to complete.
- Let \( a \) be the time (in minutes) for one arm routine.
- Let \( b \) be the time (in minutes) for one abdominal routine.
From the problem, we can create the following system of equations based on the information provided:
1. From last week:
\[ 2a + 3b = 110 \] (Equation 1)
2. From this week:
\[ 4a + 4b = 160 \] (Equation 2)
Now we can simplify Equation 2. Dividing the entire equation by 4 gives:
\[ a + b = 40 \] (Equation 3)
Now we have the following system of equations:
1. \( 2a + 3b = 110 \)
2. \( a + b = 40 \)
We can solve this system using the substitution or elimination method. Here, we'll use substitution:
From Equation 3, we can express \( b \) in terms of \( a \):
\[ b = 40 - a \]
Now, we substitute \( b \) into Equation 1:
\[ 2a + 3(40 - a) = 110 \]
Expanding the equation:
\[ 2a + 120 - 3a = 110 \]
Now, combine like terms:
\[ -a + 120 = 110 \]
Subtracting 120 from both sides gives:
\[ -a = -10 \]
Multiplying by -1, we get:
\[ a = 10 \]
Now that we have \( a \), we can substitute it back into Equation 3 to find \( b \):
\[ b = 40 - a = 40 - 10 = 30 \]
So the solutions to our variables are:
- An arm routine takes \( a = 10 \) minutes to complete.
- An abdominal routine takes \( b = 30 \) minutes to complete.
Now we fill in the blanks:
An arm routine takes **10** minutes to complete, and an abdominal routine takes **30** minutes to complete.