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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineValentina is a hairdresser. Before her lunch break, she gave 44 haircuts and colored the hair of 11 client in 197197 minutes. After lunch, she gave 22 haircuts and colored the hair of 22 clients in 226226 minutes. How long does it take for Valentina to perform each type of service, assuming the amount of time doesn't vary from client to client?\newlineIt takes Valentina _\_ 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 elimination, and fill in the blanks.\newlineValentina is a hairdresser. Before her lunch break, she gave 44 haircuts and colored the hair of 11 client in 197197 minutes. After lunch, she gave 22 haircuts and colored the hair of 22 clients in 226226 minutes. How long does it take for Valentina to perform each type of service, assuming the amount of time doesn't vary from client to client?\newlineIt takes Valentina _\_ minutes to give a haircut and _\_ minutes to color a client's hair.
  1. Define Variables: Let's define two variables: let hh be the time it takes Valentina to give a haircut, and cc be the time it takes to color a client's hair. We can then write two equations based on the information given.
  2. Write Equations: For the morning session, the equation is 4h+1c=1974h + 1c = 197 minutes.\newlineFor the afternoon session, the equation is 2h+2c=2262h + 2c = 226 minutes.\newlineNow we have a system of equations:\newline11) 4h+c=1974h + c = 197\newline22) 2h+2c=2262h + 2c = 226
  3. Use Elimination: To use elimination, we need to make the coefficients of one of the variables the same in both equations. Let's multiply the first equation by 22 to match the coefficients of cc in the second equation.\newline2×(4h+c)=2×1972\times(4h + c) = 2\times197\newlineThis gives us:\newline8h+2c=3948h + 2c = 394
  4. Substitute and Solve: Now we have a new system of equations:\newline11) 8h+2c=3948h + 2c = 394\newline22) 2h+2c=2262h + 2c = 226\newlineWe can subtract the second equation from the first to eliminate cc.\newline(8h+2c)−(2h+2c)=394−226(8h + 2c) - (2h + 2c) = 394 - 226
  5. Find h: Simplifying the subtraction, we get:\newline6h=1686h = 168\newlineNow, to find hh, we divide both sides by 66:\newlineh=1686h = \frac{168}{6}\newlineh=28h = 28\newlineSo, it takes Valentina 2828 minutes to give a haircut.
  6. Substitute for c: Now that we know hh, we can substitute it back into one of the original equations to find cc. Let's use the first equation:\newline4h+c=1974h + c = 197\newline4(28)+c=1974(28) + c = 197\newline112+c=197112 + c = 197
  7. Substitute for c: Now that we know hh, we can substitute it back into one of the original equations to find cc. Let's use the first equation:\newline4h+c=1974h + c = 197\newline4(28)+c=1974(28) + c = 197\newline112+c=197112 + c = 197Subtracting 112112 from both sides to solve for cc, we get:\newlinec=197−112c = 197 - 112\newlinec=85c = 85\newlineSo, it takes Valentina 8585 minutes to color a client's hair.

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