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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineNancy is a salon owner. Yesterday, she did 33 haircuts and colored the hair of 44 clients, charging a total of $431\$431. Today, she did 55 haircuts and colored the hair of 44 clients, charging a total of $505\$505. How much does Nancy charge for her services?\newlineNancy 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.\newlineNancy is a salon owner. Yesterday, she did 33 haircuts and colored the hair of 44 clients, charging a total of $431\$431. Today, she did 55 haircuts and colored the hair of 44 clients, charging a total of $505\$505. How much does Nancy charge for her services?\newlineNancy charges $\$_____ for a haircut and $\$_____ for a coloring.
  1. Define Equations: Let's denote the cost of a haircut as xx and the cost of a coloring as yy. We can write two equations based on the information given.\newlineFirst equation from yesterday's charges: 33 haircuts and 44 colorings for $431\$431.\newline3x+4y=4313x + 4y = 431
  2. Write Equations: Write the second equation from today's charges: 55 haircuts and 44 colorings for $505\$505.5x+4y=5055x + 4y = 505
  3. Solve System: We now have a system of equations:\newline3x+4y=4313x + 4y = 431\newline5x+4y=5055x + 4y = 505\newlineWe can solve this system using the method of elimination or substitution. Let's use elimination by subtracting the first equation from the second to eliminate yy.\newline(5x+4y)−(3x+4y)=505−431(5x + 4y) - (3x + 4y) = 505 - 431\newline5x−3x+4y−4y=505−4315x - 3x + 4y - 4y = 505 - 431\newline2x=742x = 74
  4. Elimination Method: Solve for xx, which represents the cost of a haircut.2x=742x = 74x=742x = \frac{74}{2}x=37x = 37
  5. Solve for Haircut Cost: Now that we have the cost of a haircut, we can substitute xx back into one of the original equations to find yy, the cost of a coloring. Let's use the first equation.3(37)+4y=4313(37) + 4y = 431111+4y=431111 + 4y = 4314y=431−1114y = 431 - 1114y=3204y = 320
  6. Substitute for Coloring Cost: Solve for yy.4y=3204y = 320y=3204y = \frac{320}{4}y=80y = 80
  7. Solve for Coloring Cost: We have found the cost of a haircut and the cost of a coloring.\newlineNancy charges $37\$37 for a haircut and $80\$80 for a coloring.

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