linear eqns: 38 students, 18,19,20 yrs old; ave age 18.5 yrs

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a college algebra class of 38 students at a college was made up of people who were 18,19, and 20 years of age. the average of their ages was 18.5 years. How many of each age were in class if the number of 18yrs old was eight more than the combined number of 19 and 20 years old.

How do I solve this problem? (the answer is 23 are 18 yrs old , 9 are 19 years old, and 4 that are 20 yrs old)

I did:

x + y + z= 38
8z=x+y
and I dont know anything else



Thank you for your help!
 
Re: linear equations----- variables

x = # of 18yr olds
y = # of 19yr olds
z = # of 20yr olds

1) the first one was correct:
x+y+z=38

2) x is 8 times the sum of y and z:
x = 8(y+z)

3) The average age is 18.5:
[18x+19y+20z]/38 = 18.5
 
Re: linear equations----- variables

Thank you, but what do I do next? I tried: x=8(y+z) -------->x= 8y+8z and plugged this for x in x+y+z=38 and got 9y+9z=38 -------> y +z =38/9 ----> z=38/9-y
But what do I do know?

Thank you for your help!
 
Re: linear equations----- variables

Solve each equation to be in the form: ax+by+cz=d

Then use the addition method to eliminate variables as discussed here: http://www.wtamu.edu/academic/anns/mps/ ... sthree.htm
Scroll down about 1/4 the way until you see: Solving Systems of Linear Equations in Three Variables Using the Elimination Method

Alternatively, you may use the substitution method which it appears you've started. This way tends to be longer and more confusing in my opinion.
 
Re: linear equations----- variables

now i did:

38(18x+19y+20z/38)=(18.5)38

18x+19y+20z=703 and now plug 8y +8z for x -------> 163y+164z=703 --------> y = 4.31-1.006z and x=34.48+7.048z ??? do these steps seem right?
 
x+y+z = 38
x = y+z+8, x-y-z = 8
(18x+19y+20z)/38 = 18.5, 18x+19y+20z = 703.

Use rref (reduced row echelon form)

1 1 1 38 = 1 0 0 23
1 -1 -1 8 = 0 1 0 11
18 19 20 703 = 0 0 1 4

Hence x = 23, y = 11, z = 4 , check 23+11+4 = 38
 
Hi I got it!!!!! I didn't see the correct answer in the book, I saw 9 for y instead of 11. That's why I was having so much trouble.


Thank you :D
 
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