Q. 139 - Intermediate Algebra

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?

Step-by-Step Solution

Answer

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

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

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

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.