1. ## Three variable equation

2x- y - z = 1
2x - 3y - 4z = 0
x + y - z = 4

Could someone step me through how to solve for 3 variables? The book isn't of much use to me. When I tried to solve for y, I ended up getting y = 8, which I know is wrong.

Thanks Again!

2. Originally Posted by Hapa
2x- y - z = 1
2x - 3y - 4z = 0
x + y - z = 4

Could someone step me through how to solve for 3 variables? The book isn't of much use to me. When I tried to solve for y, I ended up getting y = 8, which I know is wrong.

Thanks Again!
The trick to these is to isolate one variable in one of the equations and then substitutute that in for the variable in one of the other equations.

1) $2x - y - z = 1$
2) $2x - 3y - 4z = 0$
3) $x + y - z = 4$

Let's get a substitution value for z first using equation #3:

$x + y - z = 4$
$-z = 4 -x - y$
$z = -4 + x + y$

Now, let's plug this into equation 1:

$2x - y - (-4 + x + y) = 1$
*distribute the negative
$2x - y + 4 - x - y = 1$
*collect like terms
$x - 2y = -3$
x = -3 + 2y

Now, go back to the first one we did for z and plug what we just found for x into that equation and find the value of z in terms of y:

So our equation for z:

$z = -4 + x + y$

$z = -4 + (-3 + 2y) + y$
*distribute and collect like terms

$z = -4 - 3 + 2y + y$
z = -7 + 3y

Now you have z completely in terms of y. Next, use the two equations for x & z that are both in terms of y and solve for y in any of the equations, I'd use # 3, it's the most simple:

$x + y + z = 4$
$(-3 + 2y) + y + (-7 + 3y) = 4$
*simplify and collect like terms & solve for y

Can you take it from here?