Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Farid and his good buddy Xavier are both mechanics at a shop that does oil changes. They are in a friendly competition to see who can complete the most oil changes in one day. Farid has already finished $7$ oil changes today, and can complete more at a rate of $1$ oil change per hour. Xavier just came on shift, and can finish $2$ oil changes every hour. Sometime during the day, the friends will be tied, with the same number of oil changes completed. How long will that take? How many oil changes will Farid and Xavier each have done? $\newline$ In $\_$ hours, both men will have completed $\_$ oil changes.

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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

Farid and his good buddy Xavier are both mechanics at a shop that does oil changes. They are in a friendly competition to see who can complete the most oil changes in one day. Farid has already finished

7

oil changes today, and can complete more at a rate of

1

oil change per hour. Xavier just came on shift, and can finish

2

oil changes every hour. Sometime during the day, the friends will be tied, with the same number of oil changes completed. How long will that take? How many oil changes will Farid and Xavier each have done?

\newline

\_

hours, both men will have completed

\_

oil changes.

Define Variables: Define the variables for the system of equations. $\newline$ Let $x$ represent the number of hours that pass from the current moment, and let $y$ represent the total number of oil changes completed by each person.
Farid's Equation: Write the equation for Farid. $\newline$ Farid has already completed $7$ oil changes and can do $1$ more oil change per hour. So, his equation based on the rate of completing oil changes is: $\newline$ $y = 1x + 7$
Xavier's Equation: Write the equation for Xavier. $\newline$ Xavier is just starting and can complete $2$ oil changes per hour. His equation is: $\newline$ $y = 2x$
Set Up System: Set up the system of equations. $\newline$ The system of equations that represents the situation is: $\newline$ $y = 1x + 7$ $\newline$ $y = 2x$
Solve Using Substitution: Solve the system using substitution. $\newline$ Since both equations are equal to $y$ , set them equal to each other to find the value of $x$ : $\newline$ $1x + 7 = 2x$
Solve for x: Solve for x. $\newline$ Subtract $1x$ from both sides of the equation to isolate $x$ : $\newline$ $1x + 7 - 1x = 2x - 1x$ $\newline$ $7 = x$
Find y: Find the value of y by substituting x back into one of the original equations. $\newline$ Using Farid's equation $y = 1x + 7$ , substitute $x = 7$ : $\newline$ $y = 1(7) + 7$ $\newline$ $y = 7 + 7$ $\newline$ $y = 14$
Verify Solution: Verify the solution by substituting $x$ into Xavier's equation. $\newline$ Using Xavier's equation $y = 2x$ , substitute $x = 7$ : $\newline$ $y = 2(7)$ $\newline$ $y = 14$ $\newline$ Since this matches the value of $y$ found using Farid's equation, the solution is correct.