word problem - solving for y

[Bryantist]

Hi, I'm having some difficulty grasping one step from the following question. It is a word problem but it must be answered by solving for y. Any assistance would be appreciated.

James ran 2 miles in 12 minutes, has to run another 5 miles and wants to average 7 miles over the entire run. What will he have to run the remaining 5 miles in on average to accomplish this?

So, total run is 7 miles. 2 miles @ 6 mins/mile + 5 miles @ "y" mins/mile (to be determined). These 2 numbers added together and divided by 7 must result in 7.

2(6) + 5(y) = 7
___________
7 miles

The answer shows

7 x 12+5(y) =7 x 7
__________
7 miles

Which results in 12 + 5(y) = 49, and then 5(y)= 49 - 12 ........ and so on.

It is not clear to me as to why we are eliminating the 7 on one side and then multiplying the other side by 7. If the 7 is eliminated on one side by dividing by 7 miles, then what are we multiplying on the other side and why? Are we multiplying by 7 miles or 7 minutes?

The rest of the answer that is provided from this point on makes sense to me, its just this part that I'm a bit confused about. I find that it doesn't get answered like other 'solve for y' questions.

Thanks for your help.

[Bryantist]

Usually, you would divide to eliminate something in an equation, but in this case since the equation is divided by 7 you have to get rid of this component to isolate "y". Therefore, you remove the division by multiplying on both sides.

Thank you for clarifying this for me Police Prep