Question

On a coordinate plane, 2 lines intersect at (6, 6).
Red beads cost $1 an ounce and gold beads cost $3 an ounce. Juanita wants to purchase a 12-ounce mixture of red and gold beads that she can sell for $2 an ounce. The solution of the system shows the number of beads needed for Juanita to break even.

x + y = 12,

x + 3y = 24

How many ounces of red beads will Juanita buy to break even?
How many ounces of gold beads will she buy?

Answer 1

To solve the system of equations representing Juanita's purchase of red and gold beads, we have the following equations:

1. \( x + y = 12 \) (this represents the total ounces of beads)
2. \( x + 3y = 24 \) (this represents the cost balance, where \( x \) is the cost of red beads and \( y \) is the cost of gold beads)

We can use substitution or elimination to solve this system. Let's solve by substitution:

From equation (1), we can solve for \( x \):

\[
x = 12 - y
\]

Now, we can substitute this expression for \( x \) into equation (2):

\[
(12 - y) + 3y = 24
\]

Simplifying this:

\[
12 - y + 3y = 24
\]
\[
12 + 2y = 24
\]
\[
2y = 24 - 12
\]
\[
2y = 12
\]
\[
y = 6
\]

Now that we have \( y \), we can substitute back to find \( x \):

\[
x = 12 - y = 12 - 6 = 6
\]

Thus, Juanita will buy:
- **6 ounces of red beads** (x = 6)
- **6 ounces of gold beads** (y = 6)

In conclusion:
- Ounces of red beads: **6**
- Ounces of gold beads: **6**