Question

Read the following problem, set up a
2x2 system of equations.
Solve by Substitution/Matrices
One canned juice drink is 15% orange juice; another is 9%
orange juice. How many liters of each should be mixed
together to get 30 liters which is 13% orange juice?

Answer 1

To solve this problem, we need to set up a system of equations and solve it to find out how many liters of each type of juice should be mixed. Let's break it down step-by-step:

1. Define the Variables:
- Let $\( x \)$ be the amount of the 15% orange juice (in liters).
- Let $\( y \)$ be the amount of the 9% orange juice (in liters).

2. Set Up the Equations:
- The total volume of the mixture should be 30 liters. Therefore, we can write:
$\[ x + y = 30 \]$
- The mixture should be 13% orange juice. The total amount of orange juice contributed by both types of juice should be 13% of 30 liters. Therefore, the second equation is:
$\[ 0.15x + 0.09y = 0.13 \times 30 \]$

3. Simplify the Second Equation:
- First, calculate $\( 0.13 \times 30 \)$:
$\[ 0.13 \times 30 = 3.9 \]$
- So, our second equation becomes:
$\[ 0.15x + 0.09y = 3.9 \]$

Now we have the system of equations:
$\[ \begin{cases} x + y = 30 \\ 0.15x + 0.09y = 3.9 \end{cases} \]$

4. Solve the System of Equations:

Substitution Method:

- Solve the first equation for $\( y \)$:
$\[ y = 30 - x \]$
- Substitute $\( y \)$ in the second equation:
$\[ 0.15x + 0.09(30 - x) = 3.9 \]$

- Simplify and solve for $\( x \)$:
$\[ 0.15x + 2.7 - 0.09x = 3.9 \]$
$\[ 0.06x + 2.7 = 3.9 \]$
$\[ 0.06x = 1.2 \]$
$\[ x = \frac{1.2}{0.06} = 20 \]$

- Now, substitute $\( x = 20 \)$ back into the first equation to find $\( y \)$:
$\[ y = 30 - 20 = 10 \]$

So, the solution to the system is:
- $\( x = 20 \)$ liters of the 15% orange juice
- $\( y = 10 \)$ liters of the 9% orange juice

Therefore, you should mix 20 liters of the 15% orange juice with 10 liters of the 9% orange juice to get 30 liters of a mixture that is 13% orange juice.