# Solve the system of linear equations using matrices. 3x-2y-4=0 2y=12-x

Question
Matrices
Solve the system of linear equations using matrices.
3x-2y-4=0
2y=12-x

2021-01-26
STEP 1
System of linear equation
$$3x-2y=4$$
$$x+2y=12$$
Augmented matrix is
$$\begin{bmatrix}3 & -2 &|&4\\1 & 2 &|&12 \end{bmatrix}$$
Reduring the matrix to reduced
Row echelon form by
Row operations
$$R_1\rightarrow \frac{1}{3}R$$
$$\begin{bmatrix}1 & \frac{-2}{3} &|&\frac{4}{3}\\1 & 2 &|&12 \end{bmatrix}$$
$$R_2 \rightarrow R_2-R_1$$
$$\begin{bmatrix}1 & \frac{-2}{3} &|&\frac{4}{3}\\0 & \frac{8}{3} &|&\frac{32}{3} \end{bmatrix}$$
$$R_2 \rightarrow \frac{8}{3}R_2$$
$$\begin{bmatrix}1 & \frac{-2}{3} &|&\frac{4}{3}\\0 & 1 &|&4 \end{bmatrix}$$
Step2
$$R_1 \rightarrow R_1+\frac{2}{3}R_2$$
$$\begin{bmatrix}1 & 0 &|&4\\0 & 1 &|&4 \end{bmatrix}$$
$$\Rightarrow x =4$$
$$y=4$$
Thus solution of system is x=4 , y=4

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$$\begin{bmatrix}-1 & 0&1 \\0 & -1&0\\0&1&1 \end{bmatrix}\begin{bmatrix}x \\ y \\z \end{bmatrix}=\begin{bmatrix}-4 \\ 2\\4 \end{bmatrix}$$
$$\begin{bmatrix}-1 & 0&1 \\ 0 & -1 &0 \\ 0&1&1 \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix}=\begin{bmatrix}-4 \\ 2 \\ 4 \end{bmatrix}$$