# Write the vector form of the general solution of the given system of linear equations. 3x_1+x_2-x_3+x_4=0 2x_1+2x_2+4x_3-6x_4=0 2x_1+x_2+3x_3-x_4=0

Forms of linear equations
Write the vector form of the general solution of the given system of linear equations.
$$3x_1+x_2-x_3+x_4=0$$
$$2x_1+2x_2+4x_3-6x_4=0$$
$$2x_1+x_2+3x_3-x_4=0$$

2020-12-18
$$\begin{bmatrix}3&1&-1&1&0\\2&2&4&-6&0\\2&1&3&-1&0\end{bmatrix}$$
Write the augmented matrix of the coefficients and constants
$$\begin{bmatrix}1&0&0&2&0\\0&1&0&-5&0\\0&0&1&0&0\end{bmatrix}$$
Transform the matrix in its reduced row echelon form.
$$x_1=-2x_4$$
$$x_2=5x_4$$
$$x_3=0$$
$$x_4=x_4 \text{ free}$$
Determine the general solution
$$\begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}=x_4 \begin{bmatrix}-2\\ 5 \\ 0 \\ 1 \end{bmatrix}$$
Rewrite the solution in vector form