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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Norma is a hairdresser. Before her lunch break, she gave $1$ haircut and colored the hair of $1$ client in $106$ minutes. After lunch, she gave $1$ haircut and colored the hair of $4$ clients in $346$ minutes. How long does it take for Norma to perform each type of service, assuming the amount of time doesn't vary from client to client? $\newline$ It takes Norma $\_$ minutes to give a haircut and $\_$ minutes to color a client's hair.

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Solve a system of equations using elimination: word problems

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

Norma is a hairdresser. Before her lunch break, she gave

1

haircut and colored the hair of

1

client in

106

minutes. After lunch, she gave

1

haircut and colored the hair of

4

clients in

346

minutes. How long does it take for Norma to perform each type of service, assuming the amount of time doesn't vary from client to client?

\newline

It takes Norma

\_

minutes to give a haircut and

\_

minutes to color a client's hair.

Define variables for services: Step $1$ : Define the variables for the services Norma provides. $\newline$ Let $x$ be the time for a haircut and $y$ be the time to color a client's hair.
Write equations based on information: Step $2$ : Write the equations based on the given information. $\newline$ First scenario: $1$ haircut + $1$ coloring = $106$ minutes. $\newline$ $x + y = 106$ $\newline$ Second scenario: $1$ haircut + $4$ colorings = $346$ minutes. $\newline$ $x + 4y = 346$
Choose variable to eliminate: Step $3$ : Choose the variable to eliminate using the elimination method. $\newline$ We will eliminate $x$ by subtracting the first equation from the second.
Perform subtraction to eliminate $x$ : Step $4$ : Perform the subtraction to eliminate $x$ . $(x + 4y) - (x + y) = 346 - 106$ $x + 4y - x - y = 346 - 106$ $3y = 240$
Solve for y: Step $5$ : Solve for y. $\newline$ $3y = 240$ $\newline$ $y = \frac{240}{3}$ $\newline$ $y = 80$
Substitute value of $y$ to find $x$ : Step $6$ : Substitute the value of $y$ back into the first equation to find $x$ . $x + 80 = 106$ $x = 106 - 80$ $x = 26$

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