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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Chloe is a salon owner. Yesterday, she did $4$ haircuts and colored the hair of $3$ clients, charging a total of $\$388$ . Today, she did $4$ haircuts and colored the hair of $4$ clients, charging a total of $\$460$ . How much does Chloe charge for her services? $\newline$ Chloe charges $\$$ _ for a haircut and $\$$ _ for a coloring.

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Solve a system of equations using any method: word problems

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

\newline

Chloe is a salon owner. Yesterday, she did

4

haircuts and colored the hair of

3

clients, charging a total of

\$388

. Today, she did

4

haircuts and colored the hair of

4

clients, charging a total of

\$460

. How much does Chloe charge for her services?

\newline

Chloe charges

\$

_____ for a haircut and

\$

_____ for a coloring.

Define Variables: Let's denote the amount Chloe charges for a haircut as $x$ and for coloring as $y$ . The first equation comes from the first day's earnings: $4$ haircuts and $3$ colorings for a total of $\$388$ . Which equation represents the provided information? $4 \times \text{haircut} + 3 \times \text{coloring} = \$(388)$ $4x + 3y = 388$
First Day's Earnings: The second equation comes from the second day's earnings: $4$ haircuts and $4$ colorings for a total of $\$460$ . Which equation represents the provided information? $4 \times \text{haircut} + 4 \times \text{coloring} = \$(460)$ $4x + 4y = 460$
Second Day's Earnings: System of equations: $\newline$ $4x + 3y = 388$ $\newline$ $4x + 4y = 460$ $\newline$ Which variable should we eliminate? $\newline$ We can subtract the first equation from the second to eliminate $x$ .
Eliminate Variable: Subtract the first equation from the second to solve for $y$ . $\$4x + 4y$ - $4x + 3y$ = $460$ - $388$ \) $4x + 4y - 4x - 3y = 460 - 388$ $y = 72$
Solve for y: Now that we have the value for y, we can substitute it back into one of the original equations to solve for x. Let's use the first equation. $\newline$ $4x + 3(72) = 388$ $\newline$ $4x + 216 = 388$ $\newline$ $4x = 388 - 216$ $\newline$ $4x = 172$ $\newline$ $x = 43$
Substitute and Solve: We found: $\newline$ $x = 43$ $\newline$ $y = 72$ $\newline$ Identify the charges for a haircut and coloring. $\newline$ Chloe charges $\$43$ for a haircut and $\$72$ for a coloring.

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