Question

Mike went to a football game with his family. They bought 5 boxes of popcorn, and 3 programs and spent $41. Sean’s family went to the same game, bought 3 boxes of popcorn, and 2 programs and spent $26. How much does each program cost at the game?

$4

$5

$6

$7

Answer 1

@zzr0ck3r can you help?

Answer 2

This is a system of two linear equations. Let B be the cost of a boxe of popcorn, and P be the cost of a program.
Mike's family bought 5 boxes of popcorn, 3 programs, and spent $41

\[5B + 3P = 41\]
Sean's family bought 3 boxes of popcorn, 2 programs, and spent $26
\[3B+2P = 26\]

Now we need to solve that system of equations to find the prices for popcorn and programs. Do you know how to do that?

Answer 3

No I dont im kind of a handicap at algebra...

Answer 4

okay, I'll walk you through it. we're going to use a method called elimination, have you heard of it?

Answer 5

Yes forsure @whpalmer4

Answer 6

the idea is we want to multiply one or both of the equations by some number that will cause us to have the coefficients of one variable be equal in magnitude but opposite in number (for example, 3 and -3) so that when we add the equations together, that variable goes poof! in a puff of smoke and leaves us with an easy equation to find the other one.

Answer 7

okay I'm following

Answer 8

\[5B+3P=41\]\[3B+2P=26\]Easiest case would be if the equations already had such a pair, but they don't. Next easiest would be if we had something where the coefficient in one was a multiple of the other. No joy there either. So, we multiply the top one by the coefficient on the bottom one, and vice versa. Sort of like making a common denominator.

Do you want to eliminate B or P? Which letter do you have it in for? :-)

Answer 9

B

Answer 10

No, use P because there a P in Ralph.

Answer 11

lmao @primeralph

Answer 12

Okay. By the way, my initials are B and P :-)

So to get of B, we'll multiply the first equation by 3 (which is the coefficient of B in the second equation) and the second equation by -5 (which is -1 * the coefficient of B in the first equation).

\[3*5B + 3*3P = 3*41\]\[-5*3B + (-5)2P = -5*26\]

or
\[~~~15B ~~+ 9P = ~~~123\]\[-15B -10P = -130\]---------------------\[(15-15)B + (9-10)P = 123-130\]
\[0B -1 P = -7\]\[-1P=-7\]\[P = 7\]

So, we've got our value for P, which was the price of a program. Now let's find the price of a box of popcorn, and check our work.

Answer 13

Lol. I love it when math is fun. I love how you're answering the question while interacting. Very good my friend very good.

Answer 14

If P =7, and we know that 5B+3P=41, we can find the value of B easily enough.

\[5B + 3(7) = 41\]\[5B + 21 = 41\]What is the value of B?

Answer 15

Thankyou now I understand!! B=4

Answer 16

Yes! So let's check our answer. When we solve a system of equations, we have to check our answer in ALL of the equations. It's possible to find a set of "solutions" that only work in some of the equations, and you look like a doofus if you only check a few and it turns out you're wrong. We don't want that :-)

So going back to our original info,
"They bought 5 boxes of popcorn, and 3 programs and spent $41. Sean’s family went to the same game, bought 3 boxes of popcorn, and 2 programs and spent $26."

5 boxes of popcorn at $4 each = $20
3 programs at $7 each = $21
$20 + $21 = $41
first family checks out okay.

3 boxes of popcorn at $4 each = $12
2 programs at $7 each = $14
$12 + $14 = $26
second family checks out okay as well.

Our solution is good!

Answer 17

youre the bomb @whpalmer4

Answer 18

Thanks :-)

Answer 19

Like @Tehsh said, it's nice when you can make it be fun, and see how there might be some actual use...always my goal to show people that it isn't as hard as they might think :-)