Question

1. a group of people are collecting 20-kg and 30-kg sacks of apples to donate to a local food bank. at the end of the collection period, they donated 1,200 kg of apples and a total of 50 sacks of apples to the local food bank.

a. write a system of equations that represents the constraints in this situation. be sure to specify the variables that you use.

b. how many 20-kg sacks and how many 30-kg sacks of apples were donated to the local food bank? explain or show how you know.

(from unit 2, lesson 12.)

1. a function multiplies its inputs by \\(\frac{7}{3}\\) then subtracts 12 to get its output. use function notation to represent this function.

Explanation:

Part 7a

Step1: Define variables

Let \( x \) be the number of 20 - kg sacks and \( y \) be the number of 30 - kg sacks.

Step2: Equation for total number of sacks

The total number of sacks is 50, so \( x + y=50 \).

Step3: Equation for total weight

The weight of each 20 - kg sack is 20 kg and each 30 - kg sack is 30 kg, and the total weight is 1200 kg. So \( 20x + 30y=1200 \).

Step1: Solve the first equation for \( x \)

From \( x + y = 50 \), we get \( x=50 - y \).

Step2: Substitute into the second equation

Substitute \( x = 50 - y \) into \( 20x+30y = 1200 \). So \( 20(50 - y)+30y=1200 \).

Step3: Simplify the equation

Expand: \( 1000-20y + 30y=1200 \). Combine like terms: \( 1000 + 10y=1200 \). Subtract 1000 from both sides: \( 10y=1200 - 1000=200 \).

Step4: Solve for \( y \)

Divide both sides by 10: \( y=\frac{200}{10}=20 \).

Step5: Solve for \( x \)

Substitute \( y = 20 \) into \( x = 50 - y \), so \( x=50 - 20 = 30 \).

Step1: Recall function notation

Function notation is \( f(x) \), where \( x \) is the input.

Step2: Apply the operations

The function multiplies the input \( x \) by \(\frac{7}{3}\) (so \(\frac{7}{3}x\)) and then subtracts 12. So \( f(x)=\frac{7}{3}x-12 \).

Answer:

Let \( x \) = number of 20 - kg sacks, \( y \) = number of 30 - kg sacks. The system of equations is \(

$$\begin{cases}x + y=50\\20x + 30y=1200\end{cases}$$

\)

Part 7b