Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Melissa is going to hire a makeup artist for a fashion show and is comparing prices. Sanjay charges $\$92$ as a booking fee and an additional $\$53$ per hour. Denise charges $\$52$ per hour, plus a booking fee of $\$95$ . 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? $\newline$ If the show lasted for _ hours, the cost would always be $\$$ _.

Full solution

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

\newline

Melissa is going to hire a makeup artist for a fashion show and is comparing prices. Sanjay charges

\$92

as a booking fee and an additional

\$53

per hour. Denise charges

\$52

per hour, plus a booking fee of

\$95

. 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?

\newline

If the show lasted for _____ hours, the cost would always be

\$

_____.

Define Variables: Let's define the variables: $\newline$ Let $x$ represent the number of hours the show lasts. $\newline$ Let $C$ represent the total cost for hiring either makeup artist.
Write Equations: We can write two equations to represent the cost for each makeup artist: $\newline$ For Sanjay: $C = 92 + 53x$ $\newline$ For Denise: $C = 95 + 52x$
Set Equations Equal: To find when the costs are the same, we set the two equations equal to each other: $\newline$ $92 + 53x = 95 + 52x$
Solve for x: Now, we solve for x by subtracting $52x$ from both sides of the equation: $\newline$ $92 + 53x - 52x = 95 + 52x - 52x$ $\newline$ $92 + x = 95$
Substitute $x$ : Next, we subtract $92$ from both sides to solve for $x$ : $92 + x - 92 = 95 - 92$ $x = 3$
Calculate Total Cost: Now that we have the value for $x$ , we can substitute it back into either of the original equations to find the cost $C$ . We'll use Sanjay's equation: $\newline$ $C = 92 + 53(3)$
Calculate Total Cost: Now that we have the value for $x$ , we can substitute it back into either of the original equations to find the cost $C$ . We'll use Sanjay's equation: $\newline$ $C = 92 + 53(3)$ Calculate the total cost: $\newline$ $C = 92 + 159$ $\newline$ $C = 251$