Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Sanjay, a caterer, is investing some money in equipment and employees to help grow his business. Recently he spent $\$98$ on equipment and hired a server who makes $\$13$ per hour. Sanjay is hoping to make up these expense at the next job that is scheduled, which pays a base of $\$95$ in addition to $\$14$ per hour that the server works. In theory, this event could pay enough to cancel out Sanjay's expenditures. How long would the job have to be? How much would the job pay? $\newline$ If the job lasted _ hours, the expenditures and pay would both be $\$$ _.

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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

Sanjay, a caterer, is investing some money in equipment and employees to help grow his business. Recently he spent

\$98

on equipment and hired a server who makes

\$13

per hour. Sanjay is hoping to make up these expense at the next job that is scheduled, which pays a base of

\$95

in addition to

\$14

per hour that the server works. In theory, this event could pay enough to cancel out Sanjay's expenditures. How long would the job have to be? How much would the job pay?

\newline

If the job lasted _____ hours, the expenditures and pay would both be

\$

_____.

Define Variables: Let $x$ represent the number of hours the job lasts, and $y$ represent the total pay for the job. $\newline$ Sanjay's expenditures are for equipment and the server's hourly wage. The total expenditure is the sum of the cost of equipment and the server's wage multiplied by the number of hours worked. $\newline$ Expenditure equation: $y = 13x + 98$
Expenditure Equation: Sanjay's pay for the job is a base pay plus an hourly rate for the server's work. $\newline$ Pay equation: $y = 14x + 95$
Pay Equation: System of equations: $\newline$ $y = 13x + 98$ (Expenditure) $\newline$ $y = 14x + 95$ (Pay) $\newline$ To find the point where expenditures and pay are equal, we set the two equations equal to each other. $\newline$ $13x + 98 = 14x + 95$
System of Equations: Solve for $x$ by isolating the variable. $\newline$ Subtract $13x$ from both sides: $\newline$ $13x + 98 - 13x = 14x + 95 - 13x$ $\newline$ $98 = x + 95$ $\newline$ Now, subtract $95$ from both sides: $\newline$ $98 - 95 = x + 95 - 95$ $\newline$ $3 = x$
Solve for x: We found $x = 3$ . Now, substitute $x$ back into one of the original equations to find $y$ . $\newline$ Using the pay equation: $y = 14x + 95$ $\newline$ Substitute $3$ for $x$ : $\newline$ $y = 14(3) + 95$ $\newline$ $y = 42 + 95$ $\newline$ $y = 137$