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------------------------------------------------ Last week I spent 104 minutes doing 1 arm routine and 3 abdominal routines. This morning I completed 3 arm routines and 1 abdominal routine and spent 80 minutes.

How long does each routine last?

How do I calculate this?

To calculate the duration of each routine, let's use a system of equations based on the information you provided. Let $A$ represent the duration of an arm routine (in minutes) and $B$ represent the duration of an abdominal routine (also in minutes).

From the information given:

Last week:

- You spent 104 minutes doing 1 arm routine and 3 abdominal routines, which can be expressed as: $1A + 3B = 104$.

This morning:

- You completed 3 arm routines and 1 abdominal routine, spending 80 minutes, which can be expressed as: $3A + 1B = 80$.

Now, you have a system of two equations:

1. $1A + 3B = 104$

2. $3A + 1B = 80$

You can solve this system of equations to find the duration of each routine (A and B).

Let's use the elimination method to solve this system:

First, multiply equation (1) by 3 and equation (2) by -1 to eliminate $B$:

1. $3A + 9B = 312$

2. $-3A - B = -80$

Now, add equation (1) to equation (2) to eliminate $A$:

$(3A - 3A) + (9B - B) = (312 - 80)$

This simplifies to:

$8B = 232$

Now, divide both sides by 8 to solve for $B$:

$B = \frac{232}{8}$

$B = 29$

So, an abdominal routine lasts 29 minutes.

Now, substitute this value back into either equation (1) or (2) to solve for $A$. Let's use equation (1):

$1A + 3(29) = 104$

$A + 87 = 104$

Subtract 87 from both sides:

$A = 104 - 87$

$A = 17$

So, an arm routine lasts 17 minutes.

To summarize:

- An arm routine lasts 17 minutes.

- An abdominal routine lasts 29 minutes.