Solving system of linear equations ( 4 variables, 3 equations)

Question

Finding the solution(s) for:

$$ \begin{align*} \begin{cases} 2x_1+x_2+3x_3+2x_4 &=5\\ x_1+x_2+x_3+2x_4&=3\\ -x_2+x_3+6x_4&=3 \end{cases} \end{align*} $$

I tried using elimination to rewrite the system in row echelon form, then back substituting. I have no idea what I did wrong, or if I was doing the elimination process correctly.. but I kept getting stuck. I am confused and not really sure how to proceed, I must be awful because I've been staring at this problem for hours.

Answer 1

Adding the third row to the first one and adding the third one to the second one you obtain \begin{array}{lcl} 2x_1+4x_3+8x_4&=&8\\ x_1+2x_3+8x_4&=&6\\ -x_2+x_3+6x_4&=&3 \end{array} Now multiply the second equation by $2$ and subtract it from the first one \begin{array}{lcl} -8x_4&=&-4\\ x_1+2x_3+8x_4&=&6\\ -x_2+x_3+6x_4&=&3 \end{array} Now you know that $x_4=2$ and hence \begin{array}{lcl} x_4&=&2\\ x_1+2x_3&=&-10\\ x_3&=&-9+x_2 \end{array} Finally subtract the last one multiplied by two from the second equation \begin{array}{lcl} x_4&=&2\\ x_1&=&8-2x_2\\ x_3&=&-9+x_2 \end{array} Now it's clear that your system has an infinite number of solutions (one per each value given to $x_2$). This is not surprising since you have four variables and just three equations.

Answer 2

You can call $x_1=p$ and get the system:

$$ \begin{align*} \begin{cases} x_2+3x_3+2x_4 &=5-2p\\ x_2+x_3+2x_4&=3-p\\ -x_2+x_3+6x_4&=3 \end{cases} \end{align*} $$

If you sum the last equation at the first and second you get

$$ \begin{align*} \begin{cases} 4x_3+8x_4 &=8-2p\\ 2x_3+8x_4&=6-p\\ -x_2+x_3+6x_4&=3 \end{cases} \end{align*} $$

Now subtract the second at the first

$$ \begin{align*} \begin{cases} 2x_3 &=2-p\\ 2x_3+8x_4&=6-p\\ -x_2+x_3+6x_4&=3 \end{cases} \end{align*} $$

And solve as usual. Your solution will be a function of $p$.

Can you finish?

Answer 3

$\begin{matrix} 2& 1 &3 &2 &|&5\\ 1 &1 &1 &2 &|&3\\ 0 &-1& 1& -2&|&-1\\ 0&-1& 1& 6& |&3\\ 0& 0& 0& -8& |&-4\end{matrix}\\ x_4 = \frac 12\\ -x_2+x_3 =0\\ x_2 = x_3 = t\\ x_1 + 2t + 1 = 3\\ x_1 = 2-2t\\$

$(x_1,x_2,x_3,x_4) = (2,0,0,\frac 12) + (-2,1,1,0)t$