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

Question

Wesley is going to hire a makeup artist for a fashion show and is comparing prices. Tisha charges $90 as a booking fee and an additional $56 per hour. Felix charges $55 per hour, plus a booking fee of $92. Depending on the length of the show, the cost could end up being the same for either artist. What would the cost be? How long would the show be?

The cost would always be $
if the show lasted
hours.

Answer 1

To model the situation with a system of equations, we can define the variables and express the costs for both artists.

Let \( h \) represent the number of hours for the show.

Cost for Tisha: The total cost \( C_T \) for Tisha can be written as: \[ C_T = 90 + 56h \]

Cost for Felix: The total cost \( C_F \) for Felix can be written as: \[ C_F = 92 + 55h \]

We want to find out when the costs are equal, so we can set the two equations equal to each other: \[ 90 + 56h = 92 + 55h \]

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

1. Subtract \( 55h \) from both sides: \[ 90 + h = 92 \]

2. Subtract 90 from both sides: \[ h = 2 \]

Now that we have the number of hours, we can find out how much the cost would be by substituting \( h = 2 \) back into either of the cost equations. Let's use Tisha's cost equation:

\[ C_T = 90 + 56(2) \] \[ C_T = 90 + 112 \] \[ C_T = 202 \]

We can also verify with Felix's cost equation:

\[ C_F = 92 + 55(2) \] \[ C_F = 92 + 110 \] \[ C_F = 202 \]

Both costs are \( 202 \), which confirms our solution is correct.

Thus, the final answers are: The cost would always be $202 if the show lasted 2 hours.