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Question from Hannah, a student:

I am confused about a question on my homework, here is an example on what I need to do:

You own a sightseeing boat tour. Kids' admission costs $ \$5$, adults' admission costs $ \$8.$
During that day you get (made-up number) $ \$521$ with 71 admissions.
How many of each passenger boarded?

I tried something like
5x + 8y = 521
x + y = 71

But, I still don't understand how to solve it! Help?

Hi Hannah,

I am going to change your made-up number to $ \$523$, then if $x$ is the number of kids and y is the number of adults the equations become

\[5x + 8y = 523\]

\[x + y = 71\]

At this point multiply both sides of the second equation by $5$ so the equations become

\[5x + 8y = 523\]

\[5x + 5y = 355\]

These equations can then be rewritten

\[5x = 523 - 8y\]

\[5x = 355 - 5y\]

Since both right sides are equal to $5x$ they must be equal and hence

\[ 523 - 8y = 355 - 5y\]

Solve for $y.$

This is probably not the way your textbook or teacher would present the solution. I expect they would multiply both sides of the second equation by $5$ as I did to get

\[5x + 8y = 523\]

\[5x + 5y = 355\]

but the next step would be to "subtract the second equation from the first equation".

$5x$ minus $5x$ is $0$

$8y$ minus $3y$ is $5y$

$523$ minus $355$ is $168$

and hence subtracting the second equation from the first yields

\[3y = 168\]

Solve for y.

I hope this helps,
Penny

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