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Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.\newlineReagan is a hairdresser. Before her lunch break, she colored the hair of 11 client in 8989 minutes. After lunch, she gave 11 haircut and colored the hair of 22 clients in 193193 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?\newlineIt takes Reagan ____\_\_\_\_ minutes to give a haircut and ____\_\_\_\_ minutes to color a client's hair.

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Q. Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks.\newlineReagan is a hairdresser. Before her lunch break, she colored the hair of 11 client in 8989 minutes. After lunch, she gave 11 haircut and colored the hair of 22 clients in 193193 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?\newlineIt takes Reagan ____\_\_\_\_ minutes to give a haircut and ____\_\_\_\_ minutes to color a client's hair.
  1. Define Variables: Let xx be the time for a haircut and yy be the time for coloring hair. Reagan colors 11 client's hair in 8989 minutes. So, the equation is 1y=891y = 89.
  2. Equations Setup: After lunch, Reagan gives 11 haircut and colors 22 clients' hair in 193193 minutes. The equation is x+2y=193x + 2y = 193.
  3. Matrix Representation: We have the system of equations: y=89y = 89 and x+2y=193x + 2y = 193.
  4. Eliminate Variable: Write the system as an augmented matrix: [0amp;1amp;89 1amp;2amp;193]\left[\begin{array}{cc|c} 0 & 1 & 89 \ 1 & 2 & 193 \end{array}\right].
  5. Calculate Haircut Time: Use the first equation to eliminate xx from the second equation. But since there's no xx in the first equation, we can just use the second equation as is: x+2(89)=193x + 2(89) = 193.
  6. Calculate Coloring Time: Calculate xx: x+178=193x + 178 = 193. Subtract 178178 from both sides to get x=193−178x = 193 - 178.
  7. Calculate Coloring Time: Calculate xx: x+178=193x + 178 = 193. Subtract 178178 from both sides to get x=193−178x = 193 - 178. x=15x = 15. So, it takes Reagan 1515 minutes to give a haircut.
  8. Calculate Coloring Time: Calculate xx: x+178=193x + 178 = 193. Subtract 178178 from both sides to get x=193−178x = 193 - 178. x=15x = 15. So, it takes Reagan 1515 minutes to give a haircut. We already have y=89y = 89 from the first equation. So, it takes Reagan 8989 minutes to color a client's hair.

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