SOLUTION: The whole third variable thing confuses me. When I try and do it by substitution I always end up with this long chain that I don't know how to reduce to find the answer for the sec

Question 6156: The whole third variable thing confuses me. When I try and do it by substitution I always end up with this long chain that I don't know how to reduce to find the answer for the second variable. Thanks in advance.
Solve for X, Y and Z
22X + 5Y + 7Z = 12
10X + 3Y + 2Z = 5
9X + 2Y + 12Z = 14

Answer by glabow(165) (Show Source): You can put this solution on YOUR website!
You are looking to find one variable in terms of the others. You do this by rearranging. Rearrange the last equation to find y. (I picked that one because it looked good. Any others will work too.)
rearranges to and then to
Now substitute this value for y in the equations. The first looks like this:

Do the same substitution with the second equation.

Now do the same elimination with the two equations in x and z.
The first equation says

Substitute x in the second equation.

[resist the urge to simplify!]

[factor out 32-322]
[divide by (32-322) !]
Now substitute the value for z in

Now substitute the values for x and z in

Check in the original equations.

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