# 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]}

Question
Forms of linear equations
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
$$\displaystyle{\left[\begin{matrix}{1}&{0}&-{1}&-{2}\\{0}&{1}&{2}&{3}\end{matrix}\right]}$$

2020-11-09
The given matrix is
$$\displaystyle{B}={\left[\begin{matrix}{1}&{0}&-{1}&-{2}\\{0}&{1}&{2}&{3}\end{matrix}\right]}$$
Reduce the given augmented matrix B in to system of linear equation
Ax=b
The matrix form of the first equation is
$$\displaystyle{\left[\begin{matrix}{1}&{0}&-{1}\\{0}&{1}&{2}\end{matrix}\right]}\cdot{\left[\begin{matrix}{x}_{{1}}\\{x}_{{2}}\\{x}_{{3}}\end{matrix}\right]}={\left[\begin{matrix}-{2}\\{3}\end{matrix}\right]}$$
where
$$\displaystyle{A}={\left[\begin{matrix}{1}&{0}&-{1}\\{0}&{1}&{2}\end{matrix}\right]},{x}={\left[\begin{matrix}{x}_{{1}}\\{x}_{{2}}\\{x}_{{3}}\end{matrix}\right]},{b}={\left[\begin{matrix}-{2}\\{3}\end{matrix}\right]}$$
We use another equation to find a. general solution
$$x_1-x_3=-2 (3)$$
$$x_2+2x_3=3 (4)$$
$$x_3=x_3 (5)$$
Then
$$\displaystyle{\left({3}\right)}\Rightarrow{x}_{{1}}=-{2}+{x}_{{3}}$$
$$\displaystyle{\left({4}\right)}\Rightarrow{x}_{{2}}={3}-{2}{x}_{{3}}$$
$$\displaystyle{\left({5}\right)}\Rightarrow{x}_{{3}}={x}_{{3}}$$
In vectors form, the general solution, we obtain
$$\displaystyle{x}={\left[\begin{matrix}{x}_{{1}}\\{x}_{{2}}\\{x}_{{3}}\end{matrix}\right]}={\left[\begin{matrix}-{2}+{x}_{{3}}\\{3}-{2}{x}_{{3}}\\{x}_{{3}}\end{matrix}\right]}={\left[\begin{matrix}-{2}\\{3}\\{0}\end{matrix}\right]}+{x}_{{3}}{\left[\begin{matrix}{1}\\-{2}\\{1}\end{matrix}\right]}$$
answer $$\displaystyle{x}={\left[\begin{matrix}-{2}\\{3}\\{0}\end{matrix}\right]}+{x}_{{3}}{\left[\begin{matrix}{1}\\-{2}\\{1}\end{matrix}\right]}$$

### Relevant Questions

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
$$\displaystyle{\left[\begin{matrix}{1}&-{1}&{0}&-{2}&{0}&{0}\\{0}&{0}&{1}&{2}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\end{matrix}\right]}$$
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. $$\begin{bmatrix}1&0&-1&-2&0\\0&1&2&3&0\end{bmatrix}$$
The coefficient matrix for a system of linear differential equations of the form $$y^1=Ay$$
has the given eigenvalues and eigenspace bases. Find the general solution for the system
$$\lambda_1=2i \Rightarrow \left\{ \begin{bmatrix}1+i\\ 2-i \end{bmatrix} \right\} , \lambda_2=-2i \Rightarrow \left\{ \begin{bmatrix}1-i\\ 2+i \end{bmatrix} \right\}$$
The coefficient matrix for a system of linear differential equations of the form $$y_1=Ay$$
has the given eigenvalues and eigenspace bases. Find the general solution for the system
$$\lambda_1=3+i \Rightarrow \left\{ \begin{bmatrix}2i\\ i \end{bmatrix} \right\} , \lambda_2=3-i \Rightarrow \left\{ \begin{bmatrix}-2i\\ -i \end{bmatrix} \right\}$$

Let B be a $$(4\times3)(4\times3)$$ matrix in reduced echelon form.

a) If B has three nonzero rows, then determine the form of B.

b) Suppose that a system of 4 linear equations in 2 unknowns has augmented matrix A, where A is a $$(4\times3)(4\times3)$$ matrix row equivalent to B.

Demonstrate that the system of equations is inconsistent.

The reduced row echelon form of the augmented matrix of a system of linear equations is given. Determine whether this system of linear equations is consistent and, if so, find its general solution. Write the solution in vector form. $$\begin{bmatrix} 1 & 0 & −1 & 3 & 9\\ 0 & 1& 2 & −5 & 8\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$

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 $$c_1$$
$$\begin{cases}x_1+x_2+x_3+x_4=0\\-x_1+x_2-x_3+x_4=0\\ x_1+x_2-x_3-x_4=0\\3x_1+x_2+x_3-x_4=0 \end{cases}$$
$$X =\begin{pmatrix}1\\-1\\-1\\1\end{pmatrix}$$
For what value of the constant c is the function f continuous on $$\displaystyle{\left(−∞,+∞\right)}?$$
 $$\displaystyle{f{{\left({x}\right)}}}={\left\lbrace{\left({c}{x}^{{2}}\right)}+{4}{x},{\left({x}^{{3}}\right)}-{c}{x}\right\rbrace}{\quad\text{if}\quad}{x}{<}{5},{\quad\text{if}\quad}{x}\Rightarrow{5}$$ c=
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.
$$\displaystyle{\left\lbrace\begin{matrix}{x}+{y}={0}\\{5}{x}-{2}{y}-{2}{z}={12}\\{2}{x}+{4}{y}+{z}={5}\end{matrix}\right.}$$