Question

# Write the system of linear equations in the form Ax = b and solve this matrix equation for x. -x_{1}+x_{2}=4 -2x_{1}+x_{2}=0

Forms of linear equations
Write the system of linear equations in the form $$\displaystyle{A}{x}={b}$$ and solve this matrix equation for x. $$\displaystyle-{x}_{{{1}}}+{x}_{{{2}}}={4}$$
$$\displaystyle-{2}{x}_{{{1}}}+{x}_{{{2}}}={0}$$

2021-02-26

Step 1 We start with this system of linear equations: $$\displaystyle-{x}_{{{1}}}+{x}_{{{2}}}={4}$$
$$\displaystyle-{2}{x}_{{{1}}}+{x}_{{{2}}}={0}$$ But we want this type of set up: $$\displaystyle{A}{X}={b}$$ So, $$\begin{bmatrix}-1 & 1 \\-2 & 1 \end{bmatrix} \begin{bmatrix}x_{1} \\ x_{2} \end{bmatrix}= \begin{bmatrix}4 \\ 0 \end{bmatrix}$$ Now, let's write the LINEAR COMBINATION FORM using the COLUMN VECTORS that are the coefficients of the matrix A: $$x_{1}\begin{bmatrix}-1 \\ -2 \end{bmatrix}+ x_{2} \begin{bmatrix}1 \\ 1 \end{bmatrix}= \begin{bmatrix}4 \\ 0 \end{bmatrix}$$

Now, let's find the augmented matrix to change it to ROW ECHELON FORM using GAUSSIAN ELIMINATION. DO NOT USE GAUSS-JORDAN ELIMINATION

Step 2 $$A=\begin{bmatrix}-1 & 1 & -4 \\-2 & 1 & 0 \end{bmatrix}$$

Step 3 $$A=\begin{bmatrix}-1 & 1 & 4 \\0 & -1 & -8 \end{bmatrix}$$

Step 4 $$A=\begin{bmatrix}1 & -1 & -4 \\0 & -1 & -8 \end{bmatrix}$$

Step 5 $$A=\begin{bmatrix}1 & -1 & -4 \\0 & -1 & -8 \end{bmatrix}$$

Then, we get: $$\displaystyle{x}_{{{1}}}-{x}_{{{2}}}=-{4}$$
$$\displaystyle{x}_{{{2}}}={8}$$

Step 6

$$\displaystyle{x}_{{{1}}}-{x}_{{{2}}}=-{4}$$
$$\displaystyle{x}_{{{2}}}={8},$$
$$\displaystyle{x}_{{{1}}}-{8}=-{4}$$
$$\displaystyle{x}_{{{2}}}={8},$$ So then we get: $$\displaystyle{x}_{{{1}}}=-{4}+{8}$$
$$\displaystyle{x}_{{{1}}}={4}$$

Step 7

This is your solution: $$\displaystyle{x}_{{{1}}}={4}$$
$$\displaystyle{x}_{{{2}}}={8}$$