Basic solutions of linear equations

Question

I am struggling to find the bases of these. I have put it in the form Ax=b, however all of the examples in my notes use the formula $A_Bx_B+A_Fx_F=b$ however, this only seems to work for square matrices?

Answer 1

Can you use row reduction? (Gaussian Elimination) to convert the augmented matrix for this problem into $$\left( \begin{matrix} 1 & 2 & 1 & -2 & 10 \\ 0 & 1 & 1 & 0 & 2 \\ 0 & 0 & 1 & {\scriptstyle{}^{1}\!\!\diagup\!\!{}_{4}\;} & -2 \\ \end{matrix} \right)$$ The leading 1's in each row identify the leading variables for each column. The $x_4$ variable has no leading 1 in its column and it is called a free variable. Let $x_4=t$ and now find $x_1, x_2$ and $x_3$ in terms of t.