Solve system of linear equations in RREF

Question

I need to solve a system of linear equations using rref. I need to put my answers in the following format:

I am assuming that they are two vectors, which one has a scalar s.

Could you help me out in solving this? Idk where to start.

The equation system is:

$$ x_1 + x_2 + 2x_3 = 2, \\ 6x_1 + 5x_2 + 3x_3 = -9. $$

Thank you for your help!

Answer 1

Hint:

Write the system of equations as a matrix equation:

$$\pmatrix{1 \;\;\;1 \;\;\;2 \\ 6 \;\;\;5 \;\;\;3}\pmatrix{x_1 \\ x_2 \\ x_3}=\pmatrix{2 \\ -9}$$

and perform Gaussian elimination. If this is not enough to get you started, let me know in the comments below.

Answer 2

Your given system can be written as an augmented matrix $$ A= \left[\begin{array}{rrr|r} 1 & 1 & 2 & 2 \\ 6 & 5 & 3 & -9 \end{array}\right] $$ Row-reducing allows us to write the system in reduced row-echelon form $$ \DeclareMathOperator{rref}{rref}\rref A= \left[\begin{array}{rrr|r} 1 & 0 & -7 & -19 \\ 0 & 1 & 9 & 21 \end{array}\right] $$ This matrix corresponds to the system \begin{array}{rcrcrcrcrcrcr} x_1 &&&-&7\,x_3 &=& -19 \\ &&x_2&+&9\,x_3&=&21 \end{array} Thus all solutions to our system are of the form $$ \begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} =\begin{bmatrix}-19+7\,x_3\\21-9\,x_3\\x_3\end{bmatrix} =\begin{bmatrix}-19\\21\\0\end{bmatrix}+\begin{bmatrix}7\\-9\\1\end{bmatrix}x_3 $$