Question

write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. oscar and his good buddy hansen 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. oscar has already finished 2 oil changes today, and can complete more at a rate of 3 oil changes per hour. hansen 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 many oil changes will oscar and hansen each have done? how long will that take? both men will have completed oil changes in hours.

Explanation:

Step1: Set up the equations

Let \(x\) be the number of hours and \(y\) be the number of oil - changes.
For Oscar: \(y = 3x+2\) (initial 2 changes, 3 per hour).
For Hansen: \(y = 4x\) (0 initial changes, 4 per hour).

Step2: Substitute and solve for \(x\)

Since \(y=y\), substitute \(y\) from Hansen's equation into Oscar's.
\(4x=3x + 2\)
Subtract \(3x\) from both sides: \(4x-3x=3x + 2-3x\)
\(x = 2\)

Step3: Solve for \(y\)

Substitute \(x = 2\) into Hansen's equation \(y = 4x\).
\(y=4\times2=8\)

Answer:

Both men will have completed \(8\) oil changes in \(2\) hours.