Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Felix and his good buddy Gavin 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. Felix has already finished $6$ oil changes today, and can complete more at a rate of $1$ oil change per hour. Gavin just came on shift, and can finish $3$ 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 Felix and Gavin 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

Felix and his good buddy Gavin 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. Felix has already finished

6

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

1

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

3

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 Felix and Gavin each have done?

\newline

\_

hours, both men will have completed

\_

oil changes.

Define Variables: Let's define the variables for the number of oil changes Felix and Gavin have completed. Let $F$ represent the number of oil changes Felix has completed and $G$ represent the number of oil changes Gavin has completed. We know Felix starts with $6$ oil changes and completes $1$ more per hour, so his equation is $F = 6 + 1h$ , where $h$ is the number of hours since Felix has been working. Gavin starts with $0$ oil changes and completes $3$ per hour, so his equation is $G = 3h$ . We want to find when $F$ equals $G$ , which means they have completed the same number of oil changes.
Write Equations: Now we write the system of equations based on the information given: $\newline$ $1$ . $F = 6 + 1h$ (Felix's oil changes) $\newline$ $2$ . $G = 3h$ (Gavin's oil changes) $\newline$ We want to find the value of $h$ when $F$ equals $G$ , so we set the equations equal to each other: $\newline$ $6 + 1h = 3h$
Solve for h: To solve for h, we will subtract $1h$ from both sides of the equation to get the h terms on one side: $\newline$ $6 + 1h - 1h = 3h - 1h$ $\newline$ This simplifies to: $\newline$ $6 = 2h$
Substitute and Solve: Now we divide both sides of the equation by $2$ to solve for $h$ : $\newline$ $\frac{6}{2} = \frac{2h}{2}$ $\newline$ This gives us: $\newline$ $h = 3$
Confirm Results: We have found that $h = 3$ , which means it will take $3$ hours for Felix and Gavin to have completed the same number of oil changes. Now we need to find out how many oil changes each will have done. We can substitute $h = 3$ into either of the original equations. Let's use Felix's equation: $\newline$ $F = 6 + 1h$ $\newline$ $F = 6 + 1(3)$ $\newline$ $F = 6 + 3$ $\newline$ $F = $9$
Confirm Results: We have found that $h = 3$, which means it will take $3$ hours for Felix and Gavin to have completed the same number of oil changes. Now we need to find out how many oil changes each will have done. We can substitute $h = 3$ into either of the original equations. Let's use Felix's equation:$\newline$$F = 6 + 1h$$\newline$$F = 6 + 1(3)$$\newline$$F = 6 + 3$$\newline$$F = 9$Now let's confirm that Gavin will also have completed $9$ oil changes in $3$ hours using his equation:$\newline$$G = 3h$$\newline$$3$$0$$\newline$$3$$1$