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

Question

Riley is going to hire a makeup artist for a fashion show and is comparing prices. Charlotte charges $17 as a booking fee and an additional $93 per hour. Belle charges $81 per hour, plus a booking fee of $53. Depending on the length of the show, the cost could end up being the same for either artist. How long would the show be? What would the cost be?

If the show lasted for
hours, the cost would always be $
.

Answer 1

To set up the system of equations, we need to express the total costs for both Charlotte and Belle in terms of the number of hours \( h \).

For Charlotte: \[ C_C = 17 + 93h \]

For Belle: \[ C_B = 53 + 81h \]

We want to determine when the costs for both artists will be equal, so we set the equations equal to each other:

\[ 17 + 93h = 53 + 81h \]

Now, let's solve for \( h \) using substitution.

1. First, we can rearrange the equation to consolidate the terms involving \( h \):

\[ 93h - 81h = 53 - 17 \]

This simplifies to:

\[ 12h = 36 \]

2. Next, we divide both sides by 12:

\[ h = 3 \]

Now that we have determined the length of the show, we can plug \( h \) back into either cost equation to find the total cost. Let's use Charlotte's equation:

\[ C_C = 17 + 93 \times 3 \]

Calculating the total cost:

\[ C_C = 17 + 279 = 296 \]

Thus, if the show lasted for \( \text{3} \) hours, the cost would always be \( $\text{296} \).

The completed sentences are: If the show lasted for 3 hours, the cost would always be $296.