\(\begin{cases}x_1+3x_2-2x_4=6\\x_3+4x_4=7\\0=0\end{cases}\)

Solve for the leading entry for each individual equation. Determine thee free variables, if any. \(x_1=6-3x_2+2x_4\)

\(x_2,\ free\)

\(x_3=7-4x_4\)

\(x_4,\ free\)

Parameterize the free variables.

\(\begin{cases}x_1=6-3s+2t\\x_2=s\\x_3=7-4t\\x_4=t\end{cases}\)

Write the solution in vector form.

\(x=\begin{bmatrix}6\\0\\7\\0\end{bmatrix}+s\begin{bmatrix}-3\\1\\0\\0\end{bmatrix}+t\begin{bmatrix}2\\0\\-4\\1\end{bmatrix}\)

Result:

consistent: \(x=\begin{bmatrix}6\\0\\7\\0\end{bmatrix}+s\begin{bmatrix}-3\\1\\0\\0\end{bmatrix}+t\begin{bmatrix}2\\0\\-4\\1\end{bmatrix}\)