Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Valentina is a hairdresser. Before her lunch break, she gave $4$ haircuts and colored the hair of $1$ client in $197$ minutes. After lunch, she gave $2$ haircuts and colored the hair of $2$ clients in $226$ 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? $\newline$ It takes Valentina $\_$ minutes to give a haircut and $\_$ minutes to color a client's hair.

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Grade 8

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

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

4

haircuts and colored the hair of

1

client in

197

minutes. After lunch, she gave

2

haircuts and colored the hair of

2

clients in

226

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?

\newline

It takes Valentina

\_

minutes to give a haircut and

\_

minutes to color a client's hair.

Define Variables: Let's define two variables: let $h$ be the time it takes Valentina to give a haircut, and $c$ be the time it takes to color a client's hair. We can then write two equations based on the information given.
Write Equations: For the morning session, the equation is $4h + 1c = 197$ minutes. $\newline$ For the afternoon session, the equation is $2h + 2c = 226$ minutes. $\newline$ Now we have a system of equations: $\newline$ $1$ ) $4h + c = 197$ $\newline$ $2$ ) $2h + 2c = 226$
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 $2$ to match the coefficients of $c$ in the second equation. $\newline$ $2\times(4h + c) = 2\times197$ $\newline$ This gives us: $\newline$ $8h + 2c = 394$
Substitute and Solve: Now we have a new system of equations: $\newline$ $1$ ) $8h + 2c = 394$ $\newline$ $2$ ) $2h + 2c = 226$ $\newline$ We can subtract the second equation from the first to eliminate $c$ . $\newline$ $(8h + 2c) - (2h + 2c) = 394 - 226$
Find h: Simplifying the subtraction, we get: $\newline$ $6h = 168$ $\newline$ Now, to find $h$ , we divide both sides by $6$ : $\newline$ $h = \frac{168}{6}$ $\newline$ $h = 28$ $\newline$ So, it takes Valentina $28$ minutes to give a haircut.
Substitute for c: Now that we know $h$ , we can substitute it back into one of the original equations to find $c$ . Let's use the first equation: $\newline$ $4h + c = 197$ $\newline$ $4(28) + c = 197$ $\newline$ $112 + c = 197$
Substitute for c: Now that we know $h$ , we can substitute it back into one of the original equations to find $c$ . Let's use the first equation: $\newline$ $4h + c = 197$ $\newline$ $4(28) + c = 197$ $\newline$ $112 + c = 197$ Subtracting $112$ from both sides to solve for $c$ , we get: $\newline$ $c = 197 - 112$ $\newline$ $c = 85$ $\newline$ So, it takes Valentina $85$ minutes to color a client's hair.