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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineMaura is a salon owner. Yesterday, she did 33 haircuts and colored the hair of 55 clients, charging a total of $568\$568. Today, she did 33 haircuts and colored the hair of 22 clients, charging a total of $292\$292. How much does Maura charge for her services?\newlineMaura 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 elimination, and fill in the blanks.\newlineMaura is a salon owner. Yesterday, she did 33 haircuts and colored the hair of 55 clients, charging a total of $568\$568. Today, she did 33 haircuts and colored the hair of 22 clients, charging a total of $292\$292. How much does Maura charge for her services?\newlineMaura charges $____\$\_\_\_\_ for a haircut and $____\$\_\_\_\_ for a coloring.
  1. Define variables: Let's define two variables: let xx be the cost of a haircut and yy be the cost of coloring. We can write two equations based on the information given:\newlineFor yesterday: 3x+5y=5683x + 5y = 568\newlineFor today: 3x+2y=2923x + 2y = 292
  2. Use elimination method: To use elimination, we want to eliminate one of the variables. We can do this by multiplying the second equation by −1-1 so that when we add the two equations, the xx terms will cancel out.\newlineMultiplying the second equation by −1-1 gives us: −3x−2y=−292-3x - 2y = -292
  3. Add equations: Now we add the two equations together:\newline(3x+5y)+(−3x−2y)=568+(−292)(3x + 5y) + (-3x - 2y) = 568 + (-292)\newlineThis simplifies to:\newline3y=2763y = 276
  4. Solve for y: Next, we solve for y by dividing both sides of the equation by 33:\newline3y3=2763\frac{3y}{3} = \frac{276}{3}\newliney=92y = 92\newlineSo, Maura charges $92\$92 for coloring.
  5. Substitute yy: Now that we know the cost of coloring, we can substitute y=92y = 92 into one of the original equations to find xx. We'll use the equation for today:\newline3x+2(92)=2923x + 2(92) = 292\newline3x+184=2923x + 184 = 292
  6. Solve for x: Subtract 184184 from both sides to solve for x:\newline3x=292−1843x = 292 - 184\newline3x=1083x = 108
  7. Final result: Finally, divide both sides by 33 to find the cost of a haircut:\newlinex=1083x = \frac{108}{3}\newlinex=36x = 36\newlineSo, Maura charges $36\$36 for a haircut.

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