# Write the vector form of the general solution of the given system of linear equations. x_1+2x_2-x_3+2x_5-x_6=0

Write the vector form of the general solution of the given system of linear equations. $$\displaystyle{x}_{{1}}+{2}{x}_{{2}}-{x}_{{3}}+{2}{x}_{{5}}-{x}_{{6}}={0}$$

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Vector form given in photo:

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Rewrite the corresponding augmented matrix of the system of linear equations.

$$\begin{bmatrix}1&2&-1&0&2&-1&0 \\2&4&-2&-1&0&-5&0\\-1&-2&1&1&2&4&0\\0&0&0&1&4&3&0 \end{bmatrix}$$

Transform the matrix in its reduced row echelon form.

$$\begin{bmatrix}1&2&-1&0&2&-1&0 \\0&0&0&1&4&3&0\\0&0&0&0&0&0&0\\0&0&0&0&0&0&0 \end{bmatrix}$$

Determine the peneral solution.

$$x_1=-2x_2+x_3-2x_5+x_6$$

$$x_2=x_2$$ (free)

$$x_3=x_3$$ (free)

$$x_4= -4x_5-3x_6$$

$$x_5=x_5$$ (free)

$$x_6=x_6$$ (free)

Express the solutions in vector form.

$$\begin{bmatrix}x_1\\ x_2\\ x_3\\ x_4\\ x_5 \\x_6 \end{bmatrix}=x_2 \begin{bmatrix}-2\\ 1\\ 0\\ 0\\ 0\\ 0 \end{bmatrix}+x_3 \begin{bmatrix}1\\0\\1\\0\\0\\0 \end{bmatrix}+x_6 \begin{bmatrix}1\\0\\0\\-3\\0\\1 \end{bmatrix}$$