Question

1. one week, the music store sold 2 trumpets, 3 clarinets, and 5 violins for $1,240. the next week, they sold 3 trumpets, 1 clarinet, and 4 violins for $1,027. the following week, they sold 5 trumpets, 7 clarinets, and 2 violins for $2,091. find the cost of a trumpet, clarinet, and violin.

Explanation:

Step1: Define Variables

Let \( x \) = cost of a trumpet, \( y \) = cost of a clarinet, \( z \) = cost of a violin.
We have the system of equations:

$$
LATEXBLOCK0
$$

Step2: Eliminate \( y \) from (1) and (2)

Multiply equation (2) by 3: \( 9x + 3y + 12z = 3081 \quad (4) \)
Subtract equation (1) from (4):

$$
LATEXBLOCK1
$$

Step3: Eliminate \( y \) from (2) and (3)

Multiply equation (2) by 7: \( 21x + 7y + 28z = 7189 \quad (6) \)
Subtract equation (3) from (6):

$$
LATEXBLOCK2
$$

Step4: Solve (5) and (7) for \( x \) and \( z \)

From (5): \( x = 263 - z \)
Substitute into (7):

$$
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$$

Substitute \( z = 89 \) into (5): \( x + 89 = 263 \Rightarrow x = 174 \)

Step5: Find \( y \) using equation (2)

Substitute \( x = 174 \), \( z = 89 \) into (2):

$$
LATEXBLOCK4
$$

Answer:

Trumpet: \(\$174\), Clarinet: \(\$149\), Violin: \(\$89\)