A team t-shirt costs $3 per adult and $2 per child. On a certain day, the total number of adults (a) and children (c) who bought shirts was 100, and the total money collected was $275. Which of the following options represents the number of children and the number of adults who purchased team shirts that day, and the pair of equations that can be solved to find the numbers?

Question

mathstudent55 · Answer

You need two equations.
One equation deals with the number of people, and the other equation deals with the cost.

mathstudent55 · Answer

Let's start with the equation that deals with the number of people.
There were a adults and c children
How do you represent the sum of the numbers of adults and children using a and c?

thats_kyy · Answer

By Adding it @mathstudent55

mathstudent55 · Answer

Yes, adding what?

thats_kyy · Answer

Umm Well I Really Dont Know But Ill Try By adding How many adults and children their are @mathstudent55

mathstudent55 · Answer

Right.
The number of adults is represented by the letter a.
The number of children is represented by the letter c.
The total number of people is the number of adults plus the number of children, so the total number of people is
a + c

mathstudent55 · Answer

Ok so far?

thats_kyy · Answer

Yea @mathstudent55

mathstudent55 · Answer

We are told that the total number of people is 100. 
We know the total number of people is a + c, so we can write our first equation:
a + c = 100

thats_kyy · Answer

Okay

mathstudent55 · Answer

That was the equation that dealt with the number of people.
Now we need an equation that deals with the costs.

thats_kyy · Answer

Okay

mathstudent55 · Answer

1 adult shirt costs $3 * 1
2 adult shirts cost $3 * 2
3 adult shirts cost $3 * 3
a adult shirts cost $3 * a, or simply 3a

thats_kyy · Answer

Okay Im Getting It Now..

mathstudent55 · Answer

Now we do the same for the children's shirts.
1 child shirt costs $2 * 1
2 child shirts cost $2 * 2
3 child shirts cost $2 * 3
a child shirts cost $2 * c, or simply 2c

thats_kyy · Answer

I Just Really Need To Know how Many Children And Adults They Are

mathstudent55 · Answer

The cost of the adult shirts was 3a, and the cost of the children's shirts was 2c.
The total cost is 3a + 2c
We are told the total cost is $275, which gives us our second equation:
3a + 2c = 275

thats_kyy · Answer

okay

mathstudent55 · Answer

Now we have a system of equations:
a + c = 100
3a + 2c = 275

thats_kyy · Answer

Which Is a+c=100
3a + 2c = 275

mathstudent55 · Answer

Now we solve this system of equations to find the number of adults and the number of children.
We'll use the substitution method.
Let's solve the first equation for a.
a = 100 - c
Now we substitute 100 - c in for a in the second equation:
3(100 - c) + 2c = 275
300 - 3c + 2c = 275
300 - c = 275
-c = -25
c = 25
Now we substitute c = 25 into the first equation:
a + c = 100
a + 25 = 100
a = 75
Answer: 75 adults and 25 children

thats_kyy · Answer

OMG THANK YOUU THANK YOU SOO MUCH!!! But I Have a Couple More Im Stuck On Do You Mind Helping Me Please

thats_kyy · Answer

@mathstudent55

mathstudent55 · Answer

You're welcome.
I have another 15 minutes to help you.
Please start a new post for a new problem.
I'll go there to help you.