Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Maura is a salon owner. Yesterday, she did $3$ haircuts and colored the hair of $5$ clients, charging a total of $\$568$ . Today, she did $3$ haircuts and colored the hair of $2$ clients, charging a total of $\$292$ . How much does Maura charge for her services? $\newline$ Maura 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

Maura is a salon owner. Yesterday, she did

3

haircuts and colored the hair of

5

clients, charging a total of

\$568

. Today, she did

3

haircuts and colored the hair of

2

clients, charging a total of

\$292

. How much does Maura charge for her services?

\newline

Maura 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 information given: $\newline$ For yesterday: $3x + 5y = 568$ $\newline$ For today: $3x + 2y = 292$
Use elimination method: To use elimination, we want to eliminate one of the variables. We can do this by multiplying the second equation by $-1$ so that when we add the two equations, the $x$ terms will cancel out. $\newline$ Multiplying the second equation by $-1$ gives us: $-3x - 2y = -292$
Add equations: Now we add the two equations together: $\newline$ $(3x + 5y) + (-3x - 2y) = 568 + (-292)$ $\newline$ This simplifies to: $\newline$ $3y = 276$
Solve for y: Next, we solve for y by dividing both sides of the equation by $3$ : $\newline$ $\frac{3y}{3} = \frac{276}{3}$ $\newline$ $y = 92$ $\newline$ So, Maura charges $\$92$ for coloring.
Substitute $y$ : Now that we know the cost of coloring, we can substitute $y = 92$ into one of the original equations to find $x$ . We'll use the equation for today: $\newline$ $3x + 2(92) = 292$ $\newline$ $3x + 184 = 292$
Solve for x: Subtract $184$ from both sides to solve for x: $\newline$ $3x = 292 - 184$ $\newline$ $3x = 108$
Final result: Finally, divide both sides by $3$ to find the cost of a haircut: $\newline$ $x = \frac{108}{3}$ $\newline$ $x = 36$ $\newline$ So, Maura charges $\$36$ for a haircut.