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

Question

Irma wants to buy a charm bracelet. Vindale Fine Jewelry charges $7 per charm, plus $94 for the bracelet. Malone Jewelers, in contrast, charges $31 per charm and $22 for the bracelet. If Irma wants to add a certain number of charms to her bracelet, the cost will be the same at either jewelry shop. How many charms would that be? What would the total cost of the bracelet be?

If Irma adds
charms to her bracelet, it will cost $
at either shop.

Answer 1

Let \( x \) represent the number of charms Irma wants to buy.

The cost at Vindale Fine Jewelry can be expressed as: \[ C_V = 7x + 94 \] where \( 7x \) is the cost for the charms and \( 94 \) is the cost for the bracelet.

The cost at Malone Jewelers can be expressed as: \[ C_M = 31x + 22 \] where \( 31x \) is the cost for the charms and \( 22 \) is the cost for the bracelet.

To find how many charms would make the costs equal, we create the equation: \[ 7x + 94 = 31x + 22 \]

Now we will solve for \( x \):

1. Subtract \( 7x \) from both sides: \[ 94 = 31x - 7x + 22 \] \[ 94 = 24x + 22 \]

2. Subtract \( 22 \) from both sides: \[ 94 - 22 = 24x \] \[ 72 = 24x \]

3. Divide by \( 24 \) to isolate \( x \): \[ x = \frac{72}{24} = 3 \]

So, Irma can add 3 charms to her bracelet. Now we need to find the total cost at either shop by substituting \( x = 3 \) back into the cost equations.

Calculating the cost at Vindale Fine Jewelry: \[ C_V = 7(3) + 94 = 21 + 94 = 115 \]

Calculating the cost at Malone Jewelers: \[ C_M = 31(3) + 22 = 93 + 22 = 115 \]

Both calculations confirm that the total cost is the same.

Thus, the final answers are:

If Irma adds 3 charms to her bracelet, it will cost $115 at either shop.