Question

michael breeds chickens and ducks.
last month, he sold 50 chickens and 30 ducks for $550. this month, he sold 44 chickens and 36 ducks for $532.
how much does a chicken cost, and how much does a duck cost?
a chicken costs $\square$ and a duck costs $\square$.

Explanation:

Step1: Define Variables

Let \( x \) be the cost of a chicken (in dollars) and \( y \) be the cost of a duck (in dollars).

Step2: Set Up Equations

From last month's sales: \( 50x + 30y = 550 \) (Equation 1)
From this month's sales: \( 44x + 36y = 532 \) (Equation 2)

Step3: Simplify Equations

Simplify Equation 1 by dividing by 10: \( 5x + 3y = 55 \) (Equation 1a)
Simplify Equation 2 by dividing by 4: \( 11x + 9y = 133 \) (Equation 2a)

Step4: Eliminate a Variable

Multiply Equation 1a by 3: \( 15x + 9y = 165 \) (Equation 3)
Subtract Equation 2a from Equation 3:
\( (15x + 9y) - (11x + 9y) = 165 - 133 \)
\( 4x = 32 \)

Step5: Solve for \( x \)

Divide both sides by 4: \( x = \frac{32}{4} = 8 \)

Step6: Substitute \( x \) into Equation 1a

Substitute \( x = 8 \) into \( 5x + 3y = 55 \):
\( 5(8) + 3y = 55 \)
\( 40 + 3y = 55 \)
Subtract 40: \( 3y = 15 \)
Divide by 3: \( y = 5 \)

Answer:

A chicken costs \$8 and a duck costs \$5.