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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Nancy is a salon owner. Yesterday, she did $3$ haircuts and colored the hair of $4$ clients, charging a total of $\$431$ . Today, she did $5$ haircuts and colored the hair of $4$ clients, charging a total of $\$505$ . How much does Nancy charge for her services? $\newline$ Nancy charges $\$$ _ for a haircut and $\$$ _ for a coloring.

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Solve a system of equations using any method: word problems

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

\newline

Nancy is a salon owner. Yesterday, she did

3

haircuts and colored the hair of

4

clients, charging a total of

\$431

. Today, she did

5

haircuts and colored the hair of

4

clients, charging a total of

\$505

. How much does Nancy charge for her services?

\newline

Nancy charges

\$

_____ for a haircut and

\$

_____ for a coloring.

Define Equations: Let's denote the cost of a haircut as $x$ and the cost of a coloring as $y$ . We can write two equations based on the information given. $\newline$ First equation from yesterday's charges: $3$ haircuts and $4$ colorings for $\$431$ . $\newline$ $3x + 4y = 431$
Write Equations: Write the second equation from today's charges: $5$ haircuts and $4$ colorings for $\$505$ . $5x + 4y = 505$
Solve System: We now have a system of equations: $\newline$ $3x + 4y = 431$ $\newline$ $5x + 4y = 505$ $\newline$ We 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 $y$ . $\newline$ $(5x + 4y) - (3x + 4y) = 505 - 431$ $\newline$ $5x - 3x + 4y - 4y = 505 - 431$ $\newline$ $2x = 74$
Elimination Method: Solve for $x$ , which represents the cost of a haircut. $2x = 74$ $x = \frac{74}{2}$ $x = 37$
Solve for Haircut Cost: Now that we have the cost of a haircut, we can substitute $x$ back into one of the original equations to find $y$ , the cost of a coloring. Let's use the first equation. $3(37) + 4y = 431$ $111 + 4y = 431$ $4y = 431 - 111$ $4y = 320$
Substitute for Coloring Cost: Solve for $y$ . $4y = 320$ $y = \frac{320}{4}$ $y = 80$
Solve for Coloring Cost: We have found the cost of a haircut and the cost of a coloring. $\newline$ Nancy charges $\$37$ for a haircut and $\$80$ for a coloring.

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