Question

write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.
the owner of burlington florist is assembling flower arrangements for valentines day. this morning, she assembled 8 small arrangements and 13 large arrangements, which took her a total of 120 minutes. after lunch, she arranged 12 small arrangements and 4 large arrangements, which took 56 minutes. how long does it take to assemble each type?
the florist can assemble a small arrangement in
minutes and a large one in
minutes.

Explanation:

Step1: Define Variables

Let \( x \) be the time (in minutes) to assemble a small arrangement, and \( y \) be the time for a large one.

Step2: Form Equations

From morning: \( 8x + 13y = 120 \)
From afternoon: \( 12x + 4y = 56 \)

Step3: Eliminate a Variable

Multiply first equation by 3: \( 24x + 39y = 360 \)
Multiply second equation by 2: \( 24x + 8y = 112 \)
Subtract: \( (24x + 39y) - (24x + 8y) = 360 - 112 \)
\( 31y = 248 \) → \( y = 8 \)

Step4: Solve for \( x \)

Substitute \( y = 8 \) into \( 12x + 4y = 56 \):
\( 12x + 32 = 56 \) → \( 12x = 24 \) → \( x = 2 \)

Answer:

Small arrangement: 2 minutes, Large arrangement: 8 minutes