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=1x_1-x_2+x_3=2end{cases}

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=1x_1-x_2+x_3=2end{cases}

Question
Matrices
asked 2020-11-09
Write the system of linear equations in the form Ax = b and solve this matrix equation for x.
\(\begin{cases}x_1+x_2-3x_3=-1\\-x_1+2x_2=1\\x_1-x_2+x_3=2\end{cases}\)

Answers (1)

2020-11-10
Firstly, switch to matrix form.
\(\begin{bmatrix}1&1&-3\\-1&2&0\\1&-1&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}-1\\1\\2\end{bmatrix}\)
Now, form augmented matrix and by Gaussian climination reach row reduced echelon form.
Form augmented matrix \(\begin{bmatrix}1&1&-3&-1\\-1&2&0&1\\1&-1&1&2\end{bmatrix}\)
\(R_2+R_1\rightarrow\ and\ R_3-R_1\rightarrow R_3\) \(\begin{bmatrix}1&1&-3&-1\\0&3&-3&0\\0&-2&4&3\end{bmatrix}\)
\(R_2/3\rightarrow R_2\ \ \begin{bmatrix}1&1&-3&-1\\0&3&-3&0\\0&-2&4&3\end{bmatrix}\)
\(R_1-R_2\rightarrow R_2\ and\ R_3+2R_2\rightarrow R_3\ and\ R_3/3\rightarrow R_3 \begin{bmatrix}1&0&-2&-1\\0&1&0&1.5\\0&0&1&1.5\end{bmatrix}\)
\(R_1+2R_3\rightarrow R_1\ and\ R_2+R_3\rightarrow R_2\ \begin{bmatrix}1&0&0&2\\0&1&0&1.5\\0&0&1&1.5\end{bmatrix}\)
From here, we have solution.
\(x_1=2\\x_2=1.5\\x_3=1.5\)
Result: \(x_1=2\\x_2=1.5\\x_3=1.5\)
0

Relevant Questions

asked 2021-03-04
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 \(c_1\)
\(\begin{cases}x_1+x_2+x_3+x_4=0\\-x_1+x_2-x_3+x_4=0\\ x_1+x_2-x_3-x_4=0\\3x_1+x_2+x_3-x_4=0 \end{cases}\)
\(X =\begin{pmatrix}1\\-1\\-1\\1\end{pmatrix}\)
asked 2020-12-22
The row echelon form of a system of linear equations is given.
(a) Write the system of equations corresponding to the given matrix.
Use x, y, or x, y, z, or \(x_1,x_2,x_3, x_4\)
(b) Determine whether the system is consistent. If it is consistent, give the solution.
\(\begin{matrix}1 & 0 & 3 & 0 &1 \\ 0 & 1 & 4 & 3&2\\0&0&1&2&3\\0&0&0&0&0 \end{matrix}\)
asked 2021-03-20
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
asked 2020-12-28
Write the vector form of the general solution of the given system of linear equations.
\(x_1+2x_2-x_3=0\)
\(x_1+x_2+x_3=0\)
\(x_1+3x_2-3x_3=0\)
asked 2021-03-15

Use back-substitution to solve the system of linear equations.
\(\begin{cases}x &-y &+5z&=26\\ &\ \ \ y &+2z &=1 \\ & &\ \ \ \ \ z & =6\end{cases}\)
(x,y,z)=()

asked 2021-02-09
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.)
\(\displaystyle{x}_{{1}}+{2}{x}_{{2}}+{6}{x}_{{3}}={6}\)
\(\displaystyle{x}_{{1}}+{x}_{{2}}+{3}{x}_{{3}}={3}\)
\(\displaystyle{\left({x}_{{1}},{x}_{{2}},{x}_{{3}}\right)}=\)?
asked 2020-11-06
A small grocer finds that the monthly sales y (in $) can be approximated as a function of the amount spent advertising on the radio \(x_1\)
(in $) and the amount spent advertising in the newspaper \(x_2\) (in $) according to \(y=ax_1+bx_2+c\)
The table gives the amounts spent in advertising and the corresponding monthly sales for 3 months.
\(\begin{array}{|c|c|c|}\hline \text { Advertising, } x_{1} & \text { Advertising, } x_{2} &\text{sales, y} \\ \hline $ 2400 & {$ 800} & {$ 36,000} \\ \hline $ 2000 & {$ 500} & {$ 30,000} \\ \hline $ 3000 & {$ 1000} & {$ 44,000} \\ \hline\end{array}\)
a) Use the data to write a system of linear equations to solve for a, b, and c.
b) Use a graphing utility to find the reduced row-echelon form of the augmented matrix.
c) Write the model \(y=ax_1+bx_2+c\)
d) Predict the monthly sales if the grocer spends $250 advertising on the radio and $500 advertising in the newspaper for a given month.
asked 2021-02-21
Find values of \(\displaystyle{x}_{{1}}{\quad\text{and}\quad}{x}_{{2}}\), which satisfy this system of equation
\(\displaystyle{b}{e}{g}\in{\left\lbrace{c}{a}{s}{e}{s}\right\rbrace}{1}{x}_{{1}}-{2}{x}_{{2}}=-{16}\backslash{1}{x}_{{1}}-{1}{x}_{{2}}=-{11}{e}{n}{d}{\left\lbrace{c}{a}{s}{e}{s}\right\rbrace}\)
asked 2020-11-17
Write the system of equations in the image in matrix form
\(\displaystyle{x}'_{{1}}{\left({t}\right)}={3}{x}_{{1}}{\left({t}\right)}-{2}{x}_{{2}}{\left({t}\right)}+{e}^{{t}}{x}_{{3}}{\left({t}\right)}\)
\(\displaystyle{x}'_{{2}}{\left({t}\right)}={\sin{{\left({t}\right)}}}{x}_{{1}}{\left({t}\right)}+{\cos{{\left({t}\right)}}}{x}_{{3}}{\left({t}\right)}\)
\(\displaystyle{x}'_{{3}}{\left({t}\right)}={t}^{{2}}{x}_{{1}}{\left({t}\right)}+{t}{x}^{{2}}{\left({t}\right)}+{x}_{{3}}{\left({t}\right)}\)
asked 2020-12-17
Write the vector form of the general solution of the given system of linear equations.
\(3x_1+x_2-x_3+x_4=0\)
\(2x_1+2x_2+4x_3-6x_4=0\)
\(2x_1+x_2+3x_3-x_4=0\)
...