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
ANSWERED
asked 2021-02-25
Write the system of linear equations in the form \(\displaystyle{A}{x}={b}\) and solve this matrix equation for x. \(\displaystyle-{x}_{{{1}}}+{x}_{{{2}}}={4}\)
\(\displaystyle-{2}{x}_{{{1}}}+{x}_{{{2}}}={0}\)

Answers (1)

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: \(\displaystyle{x}_{{{1}}}={4}\)
\(\displaystyle{x}_{{{2}}}={8}\)

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