Q. 364 - Intermediate Algebra

Question

In the following exercises,translate to a system of equations and solve.

Yumi wants to make 12 cups of party mix using candies and nuts.Her budget requires the party mix tocost her \(1.29 per cup.The candies are \)2.49 per cup and nuts are $0.69 per cup.How many cups of candies and how many cups of nuts should she use?

 

Step-by-Step Solution

Answer

The number of cups of candies is about 4.And,the number of cups of nuts is about 8.

Step 1.Given information

Let x represents the number of cups of candies.

Let y represents the number of cups of nuts.

Suppose that yumi wants to make 12 cups of party mix using candies and nuts.

That means,x+y=12

Her budget requires the party mix to cost her $1.29 per cup.

The candies are $2.49 per cup and the nuts are$0.69 per cup.

That means,the trail mix is 2.49x+0.69y=1.29(12)

2.49x+0.69y=15.48

Step 2.find the number of cups of candies and nuts

Consider the system of linear equations

x+y=12

2.49x+1.29y=15.48

The object is to solve the system of linear equations by substitutions

From the first equation

$$\begin{gathered} x+y=12 \\ x=12-y \end{gathered}$$

Step 3.Substitute x = 12 - y in the second equation 2 . 49 x = 0 . 69 y = 15 . 48 and solve for y.

$$\begin{gathered} 2.49x+0.69y=15.48 \\ 2.49(12-y)+0.69y=15.48 \\ 29.88-2.49y+0.69y=15.48 \\ -1.8y=15.48-29.88 \\ -1.8y=-14.4 \\ y=\frac{-14.4}{-1.8} \\ y=8 \end{gathered}$$

Step 4.Substitute y = 8   into x = y = 12 and solve for x.

$$\begin{gathered} x+y=12 \\ x+8=12 \\ x=4 \end{gathered}$$

Step 5.Check

First substitute $x=4,y=8$ into the equation $x+y=12$.

$$\begin{gathered} x+y=12 \\ 4+8=12 \\ 12=12 \end{gathered}$$

This is true.

Substitute $x=4,y=8$ into the equation $2.49x+0.69y=15.48$.

$$\begin{gathered} 2.49x+0.69y=15.48 \\ 2.49\left(4\right)+0.69\left(8\right)=15.48 \\ 9.96+5.52=15.48 \\ 15.48=15.48 \end{gathered}$$

This is true.

Therfore,The number of cups of candies is about 4.And,the number of cups of nuts is about 8.