# Thread: simple gauss elimination prob.

1. ## simple gauss elimination prob.

For some reason I can't see how:
x+3y-z = 1
-4y + 5z = 4
can become:
x = 2-11t
y =-1+5t
z=4t

Can someone explain the step/steps between?

2. Originally Posted by prime_66
For some reason I can't see how:
x+3y-z = 1
-4y + 5z = 4
can become:
x = 2-11t
y =-1+5t
z=4t

Can someone explain the step/steps between?

$\displaystyle x\ +\ 3y\ -\ z\ =\ 1$
$\displaystyle -4y\ +\ 5z\ =\ 4$

is an under-determined system, so there will be
a free parameter $\displaystyle t$ in any solution. So we choose
$\displaystyle t$so that it maximises the convenience of solving
the system in terms of it.

The constant on the RHS of the second equation is $\displaystyle 4$, as is the
coefficient of $\displaystyle y$ on the LHS. So setting $\displaystyle z\ =\ 4t$ allows
us to divide through by $\displaystyle 4$, and solve for $\displaystyle y$ in terms of $\displaystyle t$.

Having found $\displaystyle y$ and $\displaystyle z$ in terms of $\displaystyle t$
we just substitute these into the first equation to solve for $\displaystyle x$
in terms of $\displaystyle t$.

RonL

3. thanx, I thought z had to be equal to just t. but now i understand