A system of linear equations

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Question from Alfredo, a student:

Hey i got this word problem and the directions say to set up the system of equations needed to solve them, however I was also told to solve the problem.

A student club sponsored a jazz concert and charged $3 admission for students, $5 for faculty, and $8 for the general public. The total ticket sales amounted to $2542.
Three times as many students bought tickets as faculty. The general public bought twice as many tickets as did the students. Set up the equations that determine how many
tickets were sold to each group and solve.

So far I set up the system

x=students
y=faculty
z=general public

3x+5y+8z=2542
x=3y
z=2x

Now what i need help on is solving the equation. How do i go about it using the elimination method? Please help thanks!

Hi Alfredo.

That's a really good start. Now you need to start eliminating variables until you can find the value of one of them. Here's how I would continue from where you left off:

Rewrite the equations as

3x + 5y + 8z = 2542
x - 3y = 0
2x - z = 0

Eliminate z by using the first and third equations. Multiply the third equation by 8 to get

16x - 8z = 0

Add this equation to the first equation to obtain

19x + 5y = 2542

Now eliminate y by multiplying the above equation by 3, multiplying the second equation by 5 and adding the equations. This will give you one equation to solve for x.

This system however is much easier to solve using the substitution method. Starting with the system you developed

3x + 5y + 8z = 2542
x = 3y
z = 2x

z = 2x, so z = 2 (3y) = 6y.

3x + 5y + 8z = 2542, so 3 (3y) + 5y + 8 (6y) = 2542

Find y. Use y to find x, then x to find z.

Hope this helps,
Stephen and Penny

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