# Thread: [SOLVED] Guass-Jordan find all solutions of sys of equations

1. ## [SOLVED] Guass-Jordan find all solutions of sys of equations

I have to use Gauss-Jordan elimination, put in reduced row echelon form and find all solutions to the following system of equations

2x - 3y - 8z = 7
-2x + 2z - w = -7
x - y - 3z + w = 6

I end up in RRE form

1 0 -1 0 | 3
0 1 2 0 | -1
0 0 0 1 | 3

I think

x = 3 + z
y = -1 - 2z
z = ???
w = 3

How can I solve the system?

2. There are infinitely many solutions but your RRE should be

1 0 -1 0 | 2
0 1 2 0 | -1
0 0 0 1 | 3

3. I need to be more careful with my calculations. I almost had it.

So the system has infinitely many solutions because at least one variable is a free variable and can be anything.

4. Originally Posted by thekrown
I need to be more careful with my calculations. I almost had it.

So the system has infinitely many solutions because at least one variable is a free variable and can be anything.
Yeah, to see this just set w to be any number then you can solve it as there will be 3 equations, 3 unknowns. Change w, the rest of the solutions change, change w an infinite number of times, infinite number of solutions.

Look up Diophantine equations for similar things. (not sure how they work with systems of equations though)