Question

write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.
gymnasts training for an upcoming competition are practicing their routines for balance beam and floor exercise. during morning practice, rosanne practiced her beam routine 4 times and her floor routine 10 times, which took a total of 38 minutes. during afternoon practice, she ran through her beam routine 1 time and her floor routine 2 times, which took a total of 8 minutes. how long is each routine?
the beam routine is (\boxed{quad}) minutes long and the floor exercise is (\boxed{quad}) minutes long.

Explanation:

Step1: Define Variables

Let \( b \) be the time (in minutes) for a beam routine, and \( f \) be the time (in minutes) for a floor routine.

Step2: Set Up Equations

From morning practice: \( 4b + 10f = 38 \)
From afternoon practice: \( b + 2f = 8 \)

Step3: Prepare for Elimination

Multiply the afternoon equation by 4: \( 4b + 8f = 32 \)

Step4: Subtract Equations

Subtract the new afternoon equation from the morning equation:
\( (4b + 10f) - (4b + 8f) = 38 - 32 \)
\( 2f = 6 \)

Step5: Solve for \( f \)

Divide both sides by 2: \( f = 3 \)

Step6: Solve for \( b \)

Substitute \( f = 3 \) into \( b + 2f = 8 \):
\( b + 2(3) = 8 \)
\( b + 6 = 8 \)
\( b = 2 \)

Answer:

The beam routine is \( 2 \) minutes long and the floor exercise is \( 3 \) minutes long.