Q. 141 - Intermediate Algebra

Question

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

Joseph would like to make twelve pounds of a coffee blend at a cost of $\backslash ($6 per pound. He blends Ground Chicory at $\backslash )5$ a pound with Jamaican Blue Mountain at $\$9$per pound. How much of each type of coffee should he use?

Step-by-Step Solution

Answer

Chicory coffee- 9 Pounds

Jamaican Blue-3 Pounds

Step 1. Given Information

Total coffee Joseph want to make = 12 pounds

Cost per pound= $\$6$

Total amount Joseph will invest= 12$\times$6=72$\$$

Blending cost of Ground Chicory= $\$5$

Blending cost of Jamaican Blue Mountain= $\$9$

Step 2. Formation of two linear Equations

Let number of pounds of  Jamaican Blue Mountain = $y$

Let number of pounds of  Ground Chicory= $x$

As joseph has to make 12 pounds in total so,

$x+y=12$

And if we will balance the total amount of money then,

$5x+9y=72$

Now we have 2 equations as 

$$\begin{gathered} x+y=12 \\ 5x+9y=72 \end{gathered}$$

  

Step 3. Solving the equations

From the first equation

$x+y=12$

$x=12-y$

now we will substitute this value in the second equation

$$\begin{gathered} 5x+9y=72 \\ 5\left(12-y\right)+9y=72 \\ 60-5y+9y=72 \\ 60+4y=72 \\ 4y=12 \\ y=3 \end{gathered}$$

Therefore the pounds of Jamaican Blue Mountain = 3 pounds

Now put this value in the equation 

$$\begin{gathered} x=12-y \\ x=12-3 \\ x=9 \end{gathered}$$

Therefore the pounds of Ground Chicory coffee=9 pounds.

Step 4. Check the solution

Substitute $9$ for $x$ and $3$ for $y$ in the first equation formed.

$\begin{array}{rcl}x+y& =& 12\\ 9+3& =& 12\\ 12& =& 12\end{array}$

It is a true statement.

Again, substitute the values in the second equation formed.

$\begin{array}{rcl}5x+9y& =& 72\\ 5·9+9·3& =& 72\\ 45+27& =& 72\\ 72& =& 72\end{array}$

This is also a true statement.

So the point satisfies both the equations.