Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Vijay and his good buddy Doug 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. Vijay has already finished $3$ oil changes today, and can complete more at a rate of $3$ oil changes per hour. Doug just came on shift, and can finish $4$ 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 Vijay and Doug 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

Vijay and his good buddy Doug 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. Vijay has already finished

3

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

3

oil changes per hour. Doug just came on shift, and can finish

4

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 Vijay and Doug 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 since Doug started his shift, and $y$ represent the total number of oil changes completed by each person.
Vijay's Equation: Write the equation for Vijay. $\newline$ Vijay has already completed $3$ oil changes and can do $3$ more per hour. So, his equation based on the rate of completing oil changes is: $\newline$ $y = 3x + 3$
Doug's Equation: Write the equation for Doug. $\newline$ Doug is just starting and can complete $4$ oil changes per hour. So, his equation is: $\newline$ $y = 4x$
Set Up System: Set up the system of equations. $\newline$ The system of equations representing the situation is: $\newline$ $y = 3x + 3$ $\newline$ $y = 4x$
Solve by Substitution: Solve the system using substitution. $\newline$ Since both equations are equal to $y$ , set them equal to each other to find $x$ : $\newline$ $3x + 3 = 4x$
Solve for x: Solve for x. $\newline$ Subtract $3x$ from both sides of the equation: $\newline$ $3x + 3 - 3x = 4x - 3x$ $\newline$ $3 = x$
Find $y$ : Find the value of $y$ by substituting $x$ into one of the original equations. $\newline$ Using Doug's equation $y = 4x$ and substituting $x = 3$ : $\newline$ $y = 4(3)$ $\newline$ $y = 12$
Verify Solution: Verify the solution by substituting $x$ into Vijay's equation. $\newline$ Using Vijay's equation $y = 3x + 3$ and substituting $x = 3$ : $\newline$ $y = 3(3) + 3$ $\newline$ $y = 9 + 3$ $\newline$ $y = 12$ $\newline$ Since the value of $y$ is the same for both equations, the solution is correct.