Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. Roy is going to hire a makeup artist for a fashion show and is comparin... Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. Roy is going to hire a makeup artist for a fashion show and is comparing prices. Jackie charges $29 as a booking fee and an additional $45 per hour. Emily charges $43 per hour, plus a booking fee of $37. 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.
Understand the Problem
The question is asking to create a system of equations representing the cost of hiring two makeup artists based on their pricing structures. It requires us to find out the total cost for each artist using the length of the show as a variable, and to determine when the costs equalize and how long the show would have to be.
Answer
The cost would always be $209 if the show lasted 4 hours.
Answer for screen readers
The cost would always be $209 if the show lasted 4 hours.
Steps to Solve
-
Define Variables for Costs
Let ( h ) be the length of the show in hours. -
Set Up Jackie’s Cost Equation
Jackie charges a booking fee of $29 and $45 per hour.
The cost, ( C_J ), can be expressed as:
$$ C_J = 29 + 45h $$ -
Set Up Emily’s Cost Equation
Emily charges a booking fee of $37 and $43 per hour.
The cost, ( C_E ), can be expressed as:
$$ C_E = 37 + 43h $$ -
Set the Equations Equal to Each Other
To find when their costs are the same, set the two equations equal:
$$ 29 + 45h = 37 + 43h $$ -
Solve for ( h )
Rearranging the equation:
$$ 45h - 43h = 37 - 29 $$
$$ 2h = 8 $$
Now, divide both sides by 2:
$$ h = 4 $$ -
Calculate Total Cost
Using either cost equation, plug in ( h = 4 ):
For Jackie:
$$ C_J = 29 + 45(4) = 29 + 180 = 209 $$
For Emily:
$$ C_E = 37 + 43(4) = 37 + 172 = 209 $$
The total cost for both artists for a 4-hour show is $209.
The cost would always be $209 if the show lasted 4 hours.
More Information
This scenario illustrates how two different pricing strategies can lead to the same cost under certain conditions. The solution also emphasizes the importance of setting up equations based on different pricing models to compare total costs.
Tips
- Misinterpreting the equations: Sometimes, it’s easy to mix up the charges per hour and the booking fees. Make sure you clearly define what each part of the equation represents.
- Incorrectly solving for ( h ): Be careful with the arithmetic when isolating ( h ). Double-check the steps to ensure accuracy.
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