Question

A group of children, adults, and senior citizens attended three different art exhibits that have different ticket prices for each age group. Let c represent the number of children, a represent the number of adults, and s represent the number of senior citizens. The system represents the cost of each type of ticket and the total cost of the tickets for all three exhibits. What are the number of children, adults, and seniors that attended these three exhibits?

6c + 7a + 2s = 990
3c + 3a + 4s = 430
2c + 4a + 4s = 480

A) c = 120, a = 30, s = 30
B) c = 20, a = 40, s = 70
C) c = 100, a = 50, s = 5
D) c = 100, a = 50, s = 20

Answer 1

Option D) c = 100, a = 50, s = 20

Explanation:

we have

6c + 7a + 2s = 990 -----> equation A

3c + 3a + 4s = 430 -----> equation B

2c + 4a + 4s = 480 ----> equation C

Isolate variable s in equation A

2s = 990 -6c-7a

s=495-3c-3.5a -----> equation D

Substitute equation D in equation B and in equation C

3c + 3a + 4(495-3c-3.5a) = 430

3c + 3a +1,980-12c-14a= 430

-9c-11a=-1,550 -----> equation E

2c + 4a + 4(495-3c-3.5a) = 480

2c+4a+1,980-12c-14a=480

-10c-10a=-1,500 -----> 10c+10a=1,500 -----> equation F

Solve the system of equations E and F by graphing

using a graphing tool

Let

The variable c -----> the x-axis

The variable a ----> the y-axis

The solution is the point (50,100)

see the attached figure

so

c=50, a=100

Find the value of s

s=495-3(50)-3.5(100)=-5 -----> is not make sense

therefore

Let

The variable c -----> the y-axis

The variable a ----> the x-axis

The solution is the point (50,100)

so

a=50, c=100

Find the value of s

s=495-3(100)-3.5(50)=20

therefore

The solution is

c=100,a=50,s=20