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

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

2020-12-29
$$\begin{bmatrix}1&2&-1&0\\1&1&1&0\\1&3&-3&0\end{bmatrix}$$
Write the augmented matrix of the coefficients and constants
$$\begin{bmatrix}1&0&3&0\\0&1&-2&0\\0&0&0&0\end{bmatrix}$$
Transform the matrix in its reduced row echelon form.
$$x_1=-3x_3$$
$$x_2=2x_3$$
$$x_3=x_3 \text{ free}$$
Determine the general solution
$$\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=x_4 \begin{bmatrix}-3\\ 2 \\ 1 \end{bmatrix}$$
Rewrite the solution in vector form

### Relevant Questions

Write the vector form of the general solution of the given system of linear equations.
$$3x_1+x_2-x_3+x_4=0$$
$$2x_1+2x_2+4x_3-6x_4=0$$
$$2x_1+x_2+3x_3-x_4=0$$
Write the homogeneous system of linear equations in the form AX = 0. Then verify by matrix multiplication that the given matrix X is a solution of the system for any real number $$c_1$$
$$\begin{cases}x_1+x_2+x_3+x_4=0\\-x_1+x_2-x_3+x_4=0\\ x_1+x_2-x_3-x_4=0\\3x_1+x_2+x_3-x_4=0 \end{cases}$$
$$X =\begin{pmatrix}1\\-1\\-1\\1\end{pmatrix}$$
Write the system of linear equations in the form Ax = b and solve this matrix equation for x.
$$\begin{cases}x_1+x_2-3x_3=-1\\-x_1+2x_2=1\\x_1-x_2+x_3=2\end{cases}$$
Consider a solution $$\displaystyle{x}_{{1}}$$ of the linear system Ax=b. Justify the facts stated in parts (a) and (b):
a) If $$\displaystyle{x}_{{h}}$$ is a solution of the system Ax=0, then $$\displaystyle{x}_{{1}}+{x}_{{h}}$$ is a solution of the system Ax=b.
b) If $$\displaystyle{x}_{{2}}$$ is another solution of the system Ax=b, then $$\displaystyle{x}_{{2}}-{x}_{{1}}$$ is a solution of the system Ax=0
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
$$\displaystyle{\left[\begin{matrix}{1}&{0}&-{1}&-{2}\\{0}&{1}&{2}&{3}\end{matrix}\right]}$$
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
$$\displaystyle{\left[\begin{matrix}{1}&-{1}&{0}&-{2}&{0}&{0}\\{0}&{0}&{1}&{2}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\end{matrix}\right]}$$
Solve the following system of equations. (Write your answers as a comma-separated list. If there are infinitely many solutions, write a parametric solution using t and or s. If there is no solution, write NONE.)
$$\displaystyle{x}_{{1}}+{2}{x}_{{2}}+{6}{x}_{{3}}={6}$$
$$\displaystyle{x}_{{1}}+{x}_{{2}}+{3}{x}_{{3}}={3}$$
$$\displaystyle{\left({x}_{{1}},{x}_{{2}},{x}_{{3}}\right)}=$$?
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. $$\begin{bmatrix}1&0&-1&-2&0\\0&1&2&3&0\end{bmatrix}$$
$$x=at+b , y=ct+d , z=et+f$$
$$2x_1+x_2+x_3=3$$
$$-2x_1+x_2-x_3=1$$
Use x, y, or x, y, z, or $$x_1,x_2,x_3, x_4$$
$$\begin{matrix}1 & 0 & 3 & 0 &1 \\ 0 & 1 & 4 & 3&2\\0&0&1&2&3\\0&0&0&0&0 \end{matrix}$$