$$\begin{align} &-3x_1 & &+x_3 &+x_4 &= 0\\ &x_1 &-x_2 &+2x_3 & &= 0\\ & &x_2 &-5x_3 &+2x_4 &= 0\\ &2x_1 & &+2x_3 &-3x_4 &= 0 \end{align} $$
I know the method of doing this by hand and can get the Reduce Row Echelon Form but even then I just get something like,
$x_1 = \frac{5}{8} x_4$
$x_2 = \frac{19}{8} x_4$
$x_3 = \frac{7}{8} x_4$
How do I solve this though? I know the ratios but I just can't seem to figure out what I'm missing. Thank you in advance for your help.
Your answers appear to be correct. You will not be able to find a specific answer for x4. This is because not all 4 equations are linearly independent and in this case, the 4th equation is achieved by adding together a combination of the first 3 equations. In this case, the 4th equation can be achieved: $$Equation4=-(Equation1)-(Equation2)-(Equation3)$$
Hope this helps.