Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Reagan is using ribbon to create girls' hair barrettes. For a craft fair in Allenville, she made 14 small barrettes and 12 large barrettes, using a total of 152 meters of ribbon. Then, for another craft fair in Lowell, she made 12 small barrettes and 12 large barrettes, which used a total of 144 meters. How many yards of ribbon does Reagan use for each?

Reagan uses
meters of ribbon on each small barrette and
meters on each large one.

Answer 1

Let's denote the length of ribbon used for each small barrette as \( x \) (in meters) and the length used for each large barrette as \( y \) (in meters).

From the problem, we can form the following system of equations based on the information given:

1. For the craft fair in Allenville:
\[
14x + 12y = 152
\]

2. For the craft fair in Lowell:
\[
12x + 12y = 144
\]

Now we can solve this system of equations.

First, let's simplify the second equation:
\[
12x + 12y = 144 \implies x + y = 12 \quad \text{(dividing the entire equation by 12)}
\]

Now we have a simpler system:
1. \( 14x + 12y = 152 \)
2. \( x + y = 12 \)

We can now solve the second equation for \( y \):
\[
y = 12 - x
\]

Next, we will substitute \( y \) in the first equation:
\[
14x + 12(12 - x) = 152
\]

Expanding this gives:
\[
14x + 144 - 12x = 152
\]
Combining like terms:
\[
2x + 144 = 152
\]
Subtracting 144 from both sides:
\[
2x = 8
\]
Dividing by 2:
\[
x = 4
\]

Now, substituting \( x = 4 \) back into the equation for \( y \):
\[
y = 12 - 4 = 8
\]

Thus, we've found:
- \( x = 4 \) meters (ribbon for each small barrette)
- \( y = 8 \) meters (ribbon for each large barrette)

Finally, we convert the lengths into yards (since \( 1 \) meter is approximately \( 1.09361 \) yards, we can just mention that):

Reagan uses 4 meters of ribbon on each small barrette and 8 meters on each large one.

**Final Answer:**

Reagan uses **4** meters of ribbon on each small barrette and **8** meters on each large one.