On a coordinate plane, 2 lines intersect at (6, 6).

Question

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