Question

Translate to a system of equations and solve:

Sammy has most of the ingredients he needs to make a large batch of chili. The only items he lacks are beans and ground beef. He needs a total of 20 pounds combined of beans and ground beef and has a budget of \(3 per pound. The price of beans is \)1 per pound and the price of ground beef is $5 per pound. How many pounds of beans and how many pounds of ground beef should he purchase?

Short answer

The number of pounds of beans and ground beef are 10 and 10 respectively.

Step-by-step solution

Step 1. Given Information 

The given data is that he needs a total of 20 pounds combined of beans and ground beef and has a budget of $3 per pound. The price of beans is $1 per pound and the price of ground beef is $5 per pound.

Step 2. Calculation   

Let the number of pounds of beans be x and the number of pounds of ground beef be y.

Combined cost is $3 per pound and total number of pounds are 20 pounds that is $x+y=20--\left(1\right)$

Total value of mixture can be expressed as $1x+5y=20\times 3--\left(2\right)$

Solve equation (2) and equation (1) by subtracting (1) from equation (2),

$$\begin{gathered} x+5y-x-y=60-20 \\ 4y=40 \\ y=10 \end{gathered}$$

Substitute the value of y in equation (1) to find the value of x.

$$\begin{gathered} x+10=20 \\ x=10 \end{gathered}$$