Question

	first delivery	second delivery
number of small boxes	2	5
total weight (in kilograms)	199	79

let x be the weight (in kilograms) of each large box.
let y be the weight (in kilograms) of each small box.
(a) write a system of equations that could be used to find the weight (in kilograms) of each type of box.
□x + □y = □
□x + □y = □
(b) how much does each type of box weigh (in kilograms)?
weight of each large box: □ kilograms
weight of each small box: □ kilograms

Explanation:

Part (a)

Step1: Analyze first delivery

For the first delivery, there are 12 large boxes (each of weight \( x \)) and 2 small boxes (each of weight \( y \)), with a total weight of 199 kg. So the equation is \( 12x + 2y = 199 \).

Step2: Analyze second delivery

For the second delivery, there are 3 large boxes (each of weight \( x \)) and 5 small boxes (each of weight \( y \)), with a total weight of 79 kg. So the equation is \( 3x + 5y = 79 \).

Step1: Solve the system of equations

We have the system:
\[

$$\begin{cases} 12x + 2y = 199 \quad (1)\\ 3x + 5y = 79 \quad (2) \end{cases}$$

\]
Multiply equation (2) by 4 to get: \( 12x + 20y = 316 \quad (3) \)
Subtract equation (1) from equation (3):
\( (12x + 20y) - (12x + 2y) = 316 - 199 \)
\( 18y = 117 \)
\( y = \frac{117}{18} = 6.5 \)

Step2: Substitute \( y \) into equation (2)

Substitute \( y = 6.5 \) into \( 3x + 5y = 79 \):
\( 3x + 5(6.5) = 79 \)
\( 3x + 32.5 = 79 \)
\( 3x = 79 - 32.5 = 46.5 \)
\( x = \frac{46.5}{3} = 15.5 \)

Answer:

\( 12x + 2y = 199 \)
\( 3x + 5y = 79 \)

Part (b)