1. ## Solving for variables

Given the system of equations, solve for x,y,z.

2x + y + z = 6
x + 2y + z = 12
x + y + 2z = 24

Thanks!

2. Subract the first two equations from one another

$\displaystyle 2x+y+z=16$
$\displaystyle -(x+2y+z)=-12$
$\displaystyle =x-y=4$

solve for x in the equation

$\displaystyle x=y+4$

go back and substitute for x and reapeat the process until you have completely solve the system.

Does that help?

3. ok...I plugged in x to find that y = -12 + z.
Is this correct?
Then, would z = y + 12? How will I find the numerical values of x,y, and z?

4. Make your life easier and simplify where you can.

$\displaystyle (x + y + 2z) - (x + 2y + z) = 12$

$\displaystyle z - y = 12$

$\displaystyle (x + 2y + z) - (2x + y + z) = 6$

$\displaystyle y - x = 6$

$\displaystyle (x + y + 2z) - (2x + y + z) = 18$

$\displaystyle z - x = 18$

Now just keep substituting.

$\displaystyle x + y + 2z = 24$

$\displaystyle (z - 18) + (z - 12) + 2z = 24$

$\displaystyle 4z - 30 = 24$

$\displaystyle z = \frac{27}2$