Parameter Solving System

Question

What method is used here to find the solutions? I've tried to solve it with matrix coefficients but no dice.

Is there an immediate way to get to the solutions that I'm missing?

Answer 1

I got it by the following way.

Let $x_4=t$.

Thus, from the second and third equations we obtain $x_2=-6t$ and it's enough to solve the following system: $$x_1+x_3=-4t$$ and $$x_1-2x_3=-2t,$$ which gives $$x_3=-\frac{2}{3}t$$ and $$x_1=-\frac{10}{3}t.$$

Answer 2

Since we are missing on equation to get a whole solution we have to introduce a parameter $t$. So just derive a relation between lets say $x_1, x_2, x_3$ depending on $x_4$. So that you get something of the kind $x_1=r+sx_4$ and the same respectively for $x_2$ and $x_3$. Now introduce the parameter $t$ as $x_4=t$ and you have solved the system of equations.

In your case the parameter $t=x_4$ was choosen since the solution is given by

$$\left(-\frac{10}3t,-6t,-\frac23t,\color{red}{t}\right)$$

and the relations are the following

$$\begin{align} x_1&=-\frac{10}3x_4\\ x_2&=-6x_4\\ x_3&=-\frac23x_4\\ \Rightarrow x_4&=t \end{align}$$