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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineChloe is a salon owner. Yesterday, she did 44 haircuts and colored the hair of 33 clients, charging a total of $388\$388. Today, she did 44 haircuts and colored the hair of 44 clients, charging a total of $460\$460. How much does Chloe charge for her services?\newlineChloe charges $\$_____ for a haircut and $\$_____ for a coloring.

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineChloe is a salon owner. Yesterday, she did 44 haircuts and colored the hair of 33 clients, charging a total of $388\$388. Today, she did 44 haircuts and colored the hair of 44 clients, charging a total of $460\$460. How much does Chloe charge for her services?\newlineChloe charges $\$_____ for a haircut and $\$_____ for a coloring.
  1. Define Variables: Let's denote the amount Chloe charges for a haircut as xx and for coloring as yy. The first equation comes from the first day's earnings: 44 haircuts and 33 colorings for a total of $388\$388. Which equation represents the provided information? 4×haircut+3×coloring=$(388)4 \times \text{haircut} + 3 \times \text{coloring} = \$(388) 4x+3y=3884x + 3y = 388
  2. First Day's Earnings: The second equation comes from the second day's earnings: 44 haircuts and 44 colorings for a total of $460\$460. Which equation represents the provided information? 4×haircut+4×coloring=$(460)4 \times \text{haircut} + 4 \times \text{coloring} = \$(460) 4x+4y=4604x + 4y = 460
  3. Second Day's Earnings: System of equations:\newline4x+3y=3884x + 3y = 388\newline4x+4y=4604x + 4y = 460\newlineWhich variable should we eliminate?\newlineWe can subtract the first equation from the second to eliminate xx.
  4. Eliminate Variable: Subtract the first equation from the second to solve for yy.$4x+4y\$4x + 4y - 4x+3y4x + 3y = 460460 - 388388\)4x+4y−4x−3y=460−3884x + 4y - 4x - 3y = 460 - 388y=72y = 72
  5. 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.\newline4x+3(72)=3884x + 3(72) = 388\newline4x+216=3884x + 216 = 388\newline4x=388−2164x = 388 - 216\newline4x=1724x = 172\newlinex=43x = 43
  6. Substitute and Solve: We found:\newlinex=43x = 43\newliney=72y = 72\newlineIdentify the charges for a haircut and coloring.\newlineChloe charges $43\$43 for a haircut and $72\$72 for a coloring.

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