Question

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

Hannah has to make twenty-five gallons of punch for a potluck. The punch is made of soda and fruit drink. The cost of the soda is \($1.79$. per gallon and the cost of the fruit drink is \)$2.49$per gallon. Hannah’s budget requires that the punch cost $$2.21$per gallon. How many gallons of soda and how many gallons of fruit drink does she need?

Short answer

Hannah needs $10$gallons of soda and $15$gallons of fruit mix.

Step-by-step solution

Step 1. Identify and name what we are looking for 

We need to find the amount of soda and fruit mixed required by Hannah.

Let Hannah needs $x$gallons of soda and $y$gallons of fruit mix.

Step 2. Form the equation

The punch is of $25$gallons, so soda and fruit mix combined is equal to $25$, so

$x+y=25...\left(1\right)$

The cost of soda is $$1.79$per gallon and the cost of fruit mix is $$2.49$per gallon, while the cost of punch is $$2.21$per gallon. So the second equation can be written as

$$\begin{gathered} 1.79x+2.49y=2.21\times 25 \\ 1.79x+2.49y=55.25...\left(2\right) \end{gathered}$$

Step 3. Solve using substitution

Solve the first equation for$y$

$\begin{array}{rcl}x+y& =& 25\\ x+y-x& =& 25-x\\ y& =& 25-x...\left(3\right)\end{array}$

Now using the third equation substitute $25-x$for $y$in the second equation and solve for $x$

$\begin{array}{rcl}1.79x+2.49y& =& 55.25\\ 1.79x+2.49(25-x)& =& 55.25\\ 1.79x+62.25-2.49x& =& 55.25\\ 1.79x-2.49x+62.25-62.25& =& 55.25-62.25\\ -0.7x& =& -7\\ x& =& 10\end{array}$

Step 4. Find the value of y

Substitute $10$for $x$in the third equation

$$\begin{gathered} y=25-x \\ y=25-10 \\ y=15 \end{gathered}$$

Thus the punch contains $10$gallons of soda and$15$gallons of fruit mix.

Step 5. Check the solution

Substitute $10$for $x$and $15$for $y$in the first equation formed.

$\begin{array}{rcl}x+y& =& 25\\ 10+15& =& 25\\ 25& =& 25\end{array}$

It is a true statement.

Again, substitute the values in the second equation formed.

$\begin{array}{rcl}1.79x+2.49y& =& 55.25\\ 1.79·10+2.49·15& =& 55.25\\ 17.9+37.35& =& 55.25\\ 55.25& =& 55.25\end{array}$

This is also a true statement.

So the point satisfies both the equations.