Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ Reagan is a hairdresser. Before her lunch break, she colored the hair of $1$ client in $89$ minutes. After lunch, she gave $1$ haircut and colored the hair of $2$ clients in $193$ minutes. How long does it take for Reagan to perform each type of service, assuming the amount of time doesn't vary from client to client? $\newline$ It takes Reagan $\_\_\_\_$ minutes to give a haircut and $\_\_\_\_$ minutes to color a client's hair.

Full solution

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

\newline

Reagan is a hairdresser. Before her lunch break, she colored the hair of

1

client in

89

minutes. After lunch, she gave

1

haircut and colored the hair of

2

clients in

193

minutes. How long does it take for Reagan to perform each type of service, assuming the amount of time doesn't vary from client to client?

\newline

It takes Reagan

\_\_\_\_

minutes to give a haircut and

\_\_\_\_

minutes to color a client's hair.

Define Variables: Let $x$ be the time for a haircut and $y$ be the time for coloring hair. Reagan colors $1$ client's hair in $89$ minutes. So, the equation is $1y = 89$ .
Equations Setup: After lunch, Reagan gives $1$ haircut and colors $2$ clients' hair in $193$ minutes. The equation is $x + 2y = 193$ .
Matrix Representation: We have the system of equations: $y = 89$ and $x + 2y = 193$ .
Eliminate Variable: Write the system as an augmented matrix: $\left[\begin{array}{cc|c} 0 & 1 & 89 \ 1 & 2 & 193 \end{array}\right]$ .
Calculate Haircut Time: Use the first equation to eliminate $x$ from the second equation. But since there's no $x$ in the first equation, we can just use the second equation as is: $x + 2(89) = 193$ .
Calculate Coloring Time: Calculate $x$ : $x + 178 = 193$ . Subtract $178$ from both sides to get $x = 193 - 178$ .
Calculate Coloring Time: Calculate $x$ : $x + 178 = 193$ . Subtract $178$ from both sides to get $x = 193 - 178$ . $x = 15$ . So, it takes Reagan $15$ minutes to give a haircut.
Calculate Coloring Time: Calculate $x$ : $x + 178 = 193$ . Subtract $178$ from both sides to get $x = 193 - 178$ . $x = 15$ . So, it takes Reagan $15$ minutes to give a haircut. We already have $y = 89$ from the first equation. So, it takes Reagan $89$ minutes to color a client's hair.