Question

A wholesaler is offering two different packages of custom printed T-shirts and sweatshirts to groups for fund-raisers. One package contains 10 dozen T-shirts and 14 dozen sweatshirts for $1544. The other package contains 18 dozen T-shirts and 8 dozen sweatshirts for $1472. Write and solve a system of linear equations to find the cost, in dollars, of a dozen T-shirts. a. $76 c. $48 b. $58 d. $36

Answer 1

Oooh, these are fun! We need two equations, one will represent the number of t-shirts and sweatshirts as a quantity, and the other will represent them as dollars. So here we go...

Answer 2

Actually its not even that difficult. Here...

Answer 3

Let's deal with package A first. It tells you there are 10 dozen t's. How many t's is that?

Answer 4

120

Answer 5

Good. Now do the same with the sweats. How many sweats is 14 dozen?

Answer 6

168

Answer 7

Yes. Good. Now that's out of the way. The thing here is that within package A you have 120 t's and 168 sweats that add up to $1544. The equation for that then is (I have t's as "t" and sweats as "s", ok?) 120t + 168s = 1544. That is equation one. Get that?

Answer 8

yes

Answer 9

Now let's do the second equation in the exact same way. You have 18 dozen t's in that package and 8 dozen sweats. How many t's and how many sweats, then?

Answer 10

216 & 96

Answer 11

Good! Now we have a system of equations. Keep in mind here that we have a cost PER t and a cost PER sweat, and in the end they want the cost for a DOZEN. So we will do this first and then convert at the end. No worries. Do you know how to solve systems of equations?

Answer 12

Either the substitution method or the addition method will work here. I prefer substitution. Do you have a preference?

Answer 13

sub. never did the addition method

Answer 14

Ok, then good. I prefer the sub myself. Your system is as follows:
120t + 140s = 1544
216t + 96s = 1472.
Which do you want to solve for first? The t's ok?

Answer 15

yes

Answer 16

Ok, this is a long and hairy process because it doesn't come out evenly but t = 417/360.

Answer 17

Give me a sec...I have to run outside for a minute be right back!!!! Stay with me...ok?

Answer 18

sure thing

Answer 19

Ok, I'm back now.

Answer 20

Let me check my math before i go on. I just want to make sure I'm correct.

Answer 21

ok

Answer 22

Actually, let's scrap that idea and go with the dozens of shirts. The numbers are smaller that way. Here's our new system then:

Answer 23

10t + 14s = 1544
18t + 8s = 1472

Answer 24

yes

Answer 25

Good. Now here...doing the math and solving for t in the first equation gives you
10t = 1544 - 14s...and\[t=\frac{ 1544-14s }{ 10}\]That's what we will use for substitution into the second equation. It will be tricky because of the fraction. But oh well.

Answer 26

so then what

Answer 27

Subbing t into the second equation gives you this mess:\[18(\frac{ 1544-14s }{ 10 })+8s=1472\]Now you have to distribute the 18 into that:

Answer 28

The trick here is to have a common denominator throughout the whole entire equation; that means that each term will have to be set to have a denominator of 10.

Answer 29

\[\frac{ 27792 }{ 10 }-\frac{ 252s }{ 10 }+\frac{ 80s }{ 10 }=\frac{ 14720 }{ 10 }\]Believe it or not it will be easier to get the math done this way.

Answer 30

im following continue

Answer 31

Add your "s" terms and move the 27792/10 to the other side:\[-\frac{ 13072 }{ 10 }=-\frac{ 172s }{ 10 }\]Now you can cross multiply.

Answer 32

When you do that you get -130720 = -1720s. Solving for s you get s = 76.

Answer 33

That represents the cost of a dozen sweatshirts.

Answer 34

Thank you very much IMStuck

Answer 35

But you are asked to find the cost of a dozen t-shirts, so hold on.

Answer 36

you really need to learn the addition method...also called the elimination method. Sometimes it is easier then dealing with fractions

Answer 37

thanks for the tip

Answer 38

10t + 14(76) = 1544...10t + 1064 = 1544...10t = 1544 - 1064...10t = 480...t = 48. So it costs $48 for a dozen t-shirts. That's choice c, even though it looks to me like you have a., c., b., d. Out of order? Anyway your answer is $48.

Answer 39

kelliegirl33...he said he doesn't know the addition method. If you read the whole conversation first.

Answer 40

Where he says, "sub. never did the addition method" from above. Yes it would be easier, but not if you don't know it.

Answer 41

I did read the whole conversation. That is what we are here for...to teach. If he would have wanted to learn the addition method, I would have obliged. I meant no disrespect.

Answer 42

I understand and did not mean to be short with you. I am a high school math teacher and if I am teaching my students to learn one method before another, there is a reason for it. And no...MOST of us here at this site are not teachers...we are only helpers.

Answer 43

I understand your position and did not mean to step on your toes. I am sorry

Answer 44

Np. Didn't mean anything at all. If you are here to help others understand, then you are doing an awesome thing!