Question

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

How many pounds of nuts selling for \($6$per pound and raisins selling for \)$3$per pound should Kurt combine to obtain $120$pounds of trail mix that cost him $$5$per pound?

Short answer

Kurt should combine $80$pounds of nut and$40$pounds of raisins to get the required trail mix.

Step-by-step solution

Step 1. Identify and name what we are looking for  

We need to know the quantity of nuts and raisins in the trail mix.

Let $x$represents the number of pounds of nuts in the trail mix and $y$represents the number of pounds of raisins in the trail mix.

Step 2. Form the equations

The trail mix is of $120$pounds, so the amount of nuts and raisins should be equal to $120$and thus

$x+y=120...\left(1\right)$

The cost of nuts is $$6$per pound and the cost of raisins is $$3$per pound. The trail mix is of cost $$5$per pound. So an equation can be made as

localid="1644510142198" $$\begin{gathered} 6x+3y=5\times 120 \\ 6x+3y=600...\left(2\right) \end{gathered}$$

Step 3. Solve using substitution

Solve the first equation for$y$

$\begin{array}{rcl}x+y& =& 120\\ x+y-x& =& 120-x\\ y& =& 120-x...\left(3\right)\end{array}$

Now using the third equation substitute $120-x$for $y$in the second equation and solve for $x$.

$\begin{array}{rcl}6x+3y& =& 600\\ 6x+3(120-x)& =& 600\\ 6x+360-3x& =& 600\\ 6x-3x+360-360& =& 600-360\\ 3x& =& 240\\ x& =& 80\end{array}$

Step 4. Find the value of y

Substitute $80$for $x$in the third equation

$$\begin{gathered} y=120-x \\ y=120-80 \\ y=40 \end{gathered}$$

So the trail mix contains $80$pounds of nuts and $40$pounds of raisins.

Step 5. Check the solution

Substitute $80$for $x$and $40$for $y$in the first equation formed.

$\begin{array}{rcl}x+y& =& 120\\ 80+40& =& 120\\ 120& =& 120\end{array}$

It is a true statement.

Again, substitute the values in the second equation formed.

$\begin{array}{rcl}6x+3y& =& 600\\ 6·80+3·40& =& 600\\ 480+120& =& 600\\ 600& =& 600\end{array}$

This is also a true statement.

So the point satisfies both the equations.