Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Two car owners are in need of car repairs. Harold will need to pay the mechanic $\$4$ per minute for labor, plus $\$366$ to cover the cost of new parts. Doug will need to pay $\$392$ for parts and $\$2$ per minute for labor. Depending on how long each repair takes, the two jobs might end up costing the same amount. How much time would that take? How much would Harold and Doug each have to pay? $\newline$ If the repairs took $\_\_\_\_$ minutes, Harold and Doug would each pay $\$\_\_\_\_$ .

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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

Two car owners are in need of car repairs. Harold will need to pay the mechanic

\$4

per minute for labor, plus

\$366

to cover the cost of new parts. Doug will need to pay

\$392

for parts and

\$2

per minute for labor. Depending on how long each repair takes, the two jobs might end up costing the same amount. How much time would that take? How much would Harold and Doug each have to pay?

\newline

If the repairs took

\_\_\_\_

minutes, Harold and Doug would each pay

\$\_\_\_\_

Define Variables: Let's define the variables. $\newline$ Let $x$ represent the number of minutes the labor takes. $\newline$ Let $y$ represent the total cost for the repairs.
Harold's Total Cost: Write the equation for Harold's total cost. $\newline$ Harold's cost for labor is $\$4$ per minute, and he has a parts cost of $\$366$ . $\newline$ The equation for Harold's total cost is $y = 4x + 366$ .
Doug's Total Cost: Write the equation for Doug's total cost. $\newline$ Doug's cost for labor is $\$2$ per minute, and he has a parts cost of $\$392$ . $\newline$ The equation for Doug's total cost is $y = 2x + 392$ .
Set Equations Equal: Set the two equations equal to each other to find the value of $x$ when the costs are the same. $4x + 366 = 2x + 392$
Solve for x: Solve for x by subtracting $2x$ from both sides of the equation. $\newline$ $4x - 2x + 366 = 2x - 2x + 392$ $\newline$ $2x + 366 = 392$
Isolate Variable $x$ : Subtract $366$ from both sides to isolate the variable $x$ . $2x + 366 - 366 = 392 - 366$ $2x = 26$
Divide to Solve $x$ : Divide both sides by $2$ to solve for $x$ . $\frac{2x}{2} = \frac{26}{2}$ $x = 13$
Find Total Cost: Use the value of $x$ to find the total cost $y$ for both Harold and Doug. $\newline$ We can use either equation, but let's use Harold's equation: $y = 4x + 366$ . $\newline$ $y = 4(13) + 366$ $\newline$ $y = 52 + 366$ $\newline$ $y = 418$
Check Solution: Check the solution by plugging $x$ into Doug's equation to ensure it gives the same $y$ value. $\newline$ $y = 2x + 392$ $\newline$ $y = 2(13) + 392$ $\newline$ $y = 26 + 392$ $\newline$ $y = 418$