Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Aubrey is a salon owner. Yesterday, she did $5$ haircuts and colored the hair of $1$ client, charging a total of $\$338$ . Today, she did $4$ haircuts and colored the hair of $1$ client, charging a total of $\$289$ . How much does Aubrey charge for her services? $\newline$ Aubrey charges $\$\_$ for a haircut and $\$\_$ for a coloring.

Full solution

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

\newline

Aubrey is a salon owner. Yesterday, she did

5

haircuts and colored the hair of

1

client, charging a total of

\$338

. Today, she did

4

haircuts and colored the hair of

1

client, charging a total of

\$289

. How much does Aubrey charge for her services?

\newline

Aubrey charges

\$\_

for a haircut and

\$\_

for a coloring.

Define variables: Let's define two variables: let $x$ be the cost of a haircut, and $y$ be the cost of coloring. We can write two equations based on the given information. $\newline$ From the first day: $5x + y = 338$ (Equation $1$ ) $\newline$ From the second day: $4x + y = 289$ (Equation $2$ ) $\newline$ We will use these equations to solve for $x$ and $y$ using the elimination method.
Elimination method: To eliminate $y$ , we can subtract Equation $2$ from Equation $1$ . $(5x + y) - (4x + y) = 338 - 289$ This simplifies to: $5x - 4x + y - y = 338 - 289$
Calculate cost of haircut: Simplifying the equation further, we get: $\newline$ $x = 338 - 289$ $\newline$ $x = 49$ $\newline$ So, Aubrey charges $\$49$ for a haircut.
Substitute value of x: Now that we have the value for $x$ , we can substitute it back into either Equation $1$ or Equation $2$ to find the value of $y$ . Let's use Equation $2$ . $4(49) + y = 289$ $196 + y = 289$
Calculate cost of coloring: Subtract $196$ from both sides to solve for $y$ : $\newline$ $y = 289 - 196$ $\newline$ $y = 93$ $\newline$ So, Aubrey charges $\$93$ for coloring.