Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Miss Nelson is getting married soon. She would like to hire a professional who will come to her house and do hair and makeup for the bridal party on her wedding day. She has narrowed it down to two possible choices. A woman from Riverside Salon charges an initial fee of $\$25$ and then an additional $\$13$ per hour. Alternatively, a team from Beauty by Omar offers a free consultation and then charges a higher hourly rate of $\$38$ . If it takes a certain amount of time to finish everyone's hair and makeup, Miss Nelson will end up paying the same amount either way. How much would Miss Nelson end up paying? How long would that take? $\newline$ Miss Nelson will pay $\$$ _ if it takes _ hours to complete everyone's hair and makeup.

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Algebra 1

Solve a system of equations using substitution: word problems

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

\newline

Miss Nelson is getting married soon. She would like to hire a professional who will come to her house and do hair and makeup for the bridal party on her wedding day. She has narrowed it down to two possible choices. A woman from Riverside Salon charges an initial fee of

\$25

and then an additional

\$13

per hour. Alternatively, a team from Beauty by Omar offers a free consultation and then charges a higher hourly rate of

\$38

. If it takes a certain amount of time to finish everyone's hair and makeup, Miss Nelson will end up paying the same amount either way. How much would Miss Nelson end up paying? How long would that take?

\newline

Miss Nelson will pay

\$

_____ if it takes _____ hours to complete everyone's hair and makeup.

Define Equations: Step $1$ : Define the equations for the cost of services from both providers. $\newline$ Riverside Salon: Initial fee of $\$25$ plus $\$13$ per hour. $\newline$ Equation: $y = 13x + 25$ $\newline$ Beauty by Omar: No initial fee, $\$38$ per hour. $\newline$ Equation: $y = 38x$
Set Equations Equal: Step $2$ : Set the equations equal to each other to find when the costs are the same. $13x + 25 = 38x$
Solve for x: Step $3$ : Solve for x by isolating the variable. $\newline$ $13x - 38x = -25$ $\newline$ $-25x = -25$ $\newline$ $x = 1$
Substitute $x$ into Equation: Step $4$ : Substitute $x = 1$ back into either equation to find the total cost $y$ . Using $y = 38x$ : $y = 38(1)$ $y = 38$