Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ A fashion photographer needs to hire a stylist to prepare her models. Lacey charges $\$201$ for showing up, plus $\$71$ per hour. Cindy charges $\$198$ plus $\$74$ per hour. The photographer realizes that, given the expected duration of her photo shoot, either stylist would cost her the same amount. What would the duration be? What would the cost be? $\newline$ If the shoot lasted for _ hours, either stylist would cost $\$$ _.

View step-by-step help

Home

Math Problems

Algebra 2

Solve a system of equations using substitution: word problems

Full solution

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

\newline

A fashion photographer needs to hire a stylist to prepare her models. Lacey charges

\$201

for showing up, plus

\$71

per hour. Cindy charges

\$198

plus

\$74

per hour. The photographer realizes that, given the expected duration of her photo shoot, either stylist would cost her the same amount. What would the duration be? What would the cost be?

\newline

If the shoot lasted for _____ hours, either stylist would cost

\$

_____.

Stylist Cost Equations: Let $x$ represent the duration of the photo shoot in hours, and $y$ represent the total cost for hiring a stylist. $\newline$ For Lacey: $\newline$ Base charge: $\$201$ $\newline$ Hourly rate: $\$71$ $\newline$ Equation based on the given information: $\newline$ Total cost $y$ = base charge + (hourly rate $*$ duration $x$ ) $\newline$ $y = 71x + 201$ $\newline$ For Lacey, the equation is: $y = 71x + 201$
System of Equations: For Cindy: $\newline$ Base charge: $\$198$ $\newline$ Hourly rate: $\$74$ $\newline$ Equation based on the given information: $\newline$ Total cost $y$ = base charge + (hourly rate $*$ duration $x$ ) $\newline$ $y = 74x + 198$ $\newline$ For Cindy, the equation is: $y = 74x + 198$
Solving for Duration: System of equations: $\newline$ $y = 71x + 201$ $\newline$ $y = 74x + 198$ $\newline$ To find the duration $x$ when the cost $y$ is the same for both stylists, set the two equations equal to each other: $\newline$ $71x + 201 = 74x + 198$
Substitute and Calculate: Solve for $x$ by isolating the variable. $\newline$ Subtract $71x$ from both sides of the equation: $\newline$ $71x + 201 - 71x = 74x + 198 - 71x$ $\newline$ $201 = 3x + 198$ $\newline$ Now, subtract $198$ from both sides: $\newline$ $201 - 198 = 3x + 198 - 198$ $\newline$ $3 = 3x$
Final Result: Divide both sides by $3$ to solve for $x$ : $\frac{3}{3} = \frac{3x}{3}$ $1 = x$ So, $x = 1$
Final Result: Divide both sides by $3$ to solve for $x$ : $\frac{3}{3} = \frac{3x}{3}$ $1 = x$ So, $x = 1$ Now that we have $x = 1$ , we can find the total cost $y$ by substituting $x$ into one of the original equations. $\newline$ Substitute $1$ for $x$ in $y = 71x + 201$ : $y = 71(1) + 201$ $y = 71 + 201$ $y = 272$ So, $y = 272$
Final Result: Divide both sides by $3$ to solve for $x$ : $\frac{3}{3} = \frac{3x}{3}$ $1 = x$ So, $x = 1$ Now that we have $x = 1$ , we can find the total cost $y$ by substituting $x$ into one of the original equations. Substitute $1$ for $x$ in $y = 71x + 201$ : $y = 71(1) + 201$ $y = 71 + 201$ $y = 272$ So, $y = 272$ We have found that the duration $x$ is $1$ hour and the total cost $y$ is $x = 1$ $2$ . Therefore, if the shoot lasted for $1$ hour, either stylist would cost $x = 1$ $2$ .