# Thread: Gaussian elimination (having trouble getting matrix into row echelon form)

1. ## Gaussian elimination (having trouble getting matrix into row echelon form)

I am having a lot fo trouble with this question

Use gaussian elimination to obtain an equivalent system whose coefficient matrix s in row echelon form. If system is consistent and there are free variables, transform it to reduce row echelon form and find al solutions.

x1 + 3x2 + x3 + x4 = 3
2x1 - 2x2 + x3 + 2x4 = 8
x1 - 3x2 + x4 = 5

I reduced it as far as

1 -5 0 1 5
0 -8 -1 0 2
0 0 0 0 0

But I cant seem to get it into row echelon form. I tried dividing the 2nd row in half and subtracting (-1) of the top row.

but then I get this

1 -5 0 1 5
-1 1 -1/2 -1 -3
0 0 0 0 0

This is my first week taking linear algebra so I guess its expected to run into trouble like this.

2. Perform the row operation

$\displaystyle R1'=R1-\frac{5}{8}R2$.

3. thanks, I need to start trying fractions in these problems.

once I do this and reduce it further I get

1 0 5/8 1 15/4
0 1 1/8 0 -1/4

so I am assuming I start solving by doing this

x2 + 1/8x3 = -1/4 --> x2 = -1/4 -1/8x3

but now I'm stuck. I cant plug x2 into the first row since there is 0 x2. Did I start this incorrectly?

4. x3 is a "free variable" in this system. x1 and x2 are "dependent" variables. this is to say, you have to pick a value for x3 before you know what x1 and x2 have to be.

your answer should be of the form (a+bt, c+dt, t), the solution space is 1 dimensional (a line in 3-space), rather than 0-dimensional (a single point).

5. Look at some examples in your linear algebra book to see how they handle free variables.