# Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations. displaystyle{leftlbracebegin{matrix}{x}+{y}={0}{5}{x}-{2}{y}-{2}{z}={12}{2}{x}+{4}{y}+{z}={5}end{matrix}right.}

Question
Forms of linear equations
Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations.
$$\displaystyle{\left\lbrace\begin{matrix}{x}+{y}={0}\\{5}{x}-{2}{y}-{2}{z}={12}\\{2}{x}+{4}{y}+{z}={5}\end{matrix}\right.}$$

2021-03-05
For a system of equations $$\displaystyle{\left\lbrace\begin{matrix}{a}{x}+{b}{y}+{c}{z}={j}\\{\left.{d}{x}\right.}+{e}{y}+{f}{z}={k}\\{g}{x}+{h}{y}+{i}{z}={l}\end{matrix}\right.}$$
, the coefficient matrix is $$\displaystyle{\left[\begin{matrix}{a}&{b}&{c}\\{d}&{e}&{f}\\{g}&{h}&{i}\end{matrix}\right]}$$ and the augmented matrix is $$\displaystyle{\left[\begin{matrix}{a}&{b}&{c}&{|}&{j}\\{d}&{e}&{f}&{|}&{k}\\{g}&{h}&{i}&{|}&{l}\end{matrix}\right]}$$
a) For the system $$\displaystyle{\left\lbrace\begin{matrix}{x}+{y}={0}\\{5}{x}-{2}{y}-{2}{z}={12}\\{2}{x}+{4}{y}+{z}={5}\end{matrix}\right.}$$
a=1 , b=1 , c=0,d=5,e=-2, f=-2, g=2 , h=4 and i=1 so the coefficient matrix is $$\displaystyle{\left[\begin{matrix}{5}&{1}&{0}\\{5}&-{2}&-{2}\\{2}&{4}&{1}\end{matrix}\right]}$$
b) For the system $$\displaystyle{\left\lbrace\begin{matrix}{x}+{y}={0}\\{5}{x}-{2}{y}-{2}{z}={12}\\{2}{x}+{4}{y}+{z}={5}\end{matrix}\right.}$$
a=1 , b=1 , c=0,d=5,e=-2, f=-2, g=2 , h=4 , i=1 , j=0 , k=12 and l=5 so the augmented matrix is $$\displaystyle{\left[\begin{matrix}{5}&{1}&{0}&{|}&{0}\\{5}&-{2}&-{2}&{|}&{12}\\{2}&{4}&{1}&{|}&{5}\end{matrix}\right]}$$
a) $$\displaystyle{\left[\begin{matrix}{5}&{1}&{0}\\{5}&-{2}&-{2}\\{2}&{4}&{1}\end{matrix}\right]}$$
b) $$\displaystyle{\left[\begin{matrix}{5}&{1}&{0}&{|}&{0}\\{5}&-{2}&-{2}&{|}&{12}\\{2}&{4}&{1}&{|}&{5}\end{matrix}\right]}$$

### Relevant Questions

Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations. $$\begin{cases}8x+3y=25 \\ 3x-9y=12 \end{cases}$$
Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations $$\begin{cases}9x-3y+z=13 \\ 12x-8z=5 \\ 3x+4y-z =6 \end{cases}$$
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
$$\displaystyle{\left[\begin{matrix}{1}&{0}&-{1}&-{2}\\{0}&{1}&{2}&{3}\end{matrix}\right]}$$
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
$$\displaystyle{\left[\begin{matrix}{1}&-{1}&{0}&-{2}&{0}&{0}\\{0}&{0}&{1}&{2}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\end{matrix}\right]}$$
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 2020-10-19 The purchase price of a home y (in$1000) can be approximated based on the annual income of the buyer $$x_1$$ (in $1000) and on the square footage of the home $$x_2 (\text{ in } 100ft^2)$$ according to $$y=ax_1+bx_2+c$$ The table gives the incomes of three buyers, the square footages of the home purchased, and the corresponding purchase prices of the home. 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 purchase price for a buyer who makes$100000 per year and wants a $$2500ft^2$$ home.
The coefficient matrix for a system of linear differential equations of the form $$y^1=Ay$$
=Ay has the given eigenvalues and eigenspace bases. Find the general solution for the system \lambda_1=3 \Rightarrow \left\{ \begin{bmatrix}1\\1\\0 \end{bmatrix} \right\} , \lambda_2=0 \Rightarrow \left\{ \begin{bmatrix}1\\5\\1 \end{bmatrix} , \begin{bmatrix}2\\1\\4 \end{bmatrix} \right\}
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. $$\displaystyle{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}&−{1}&{3}&{9}\backslash{0}&{1}&{2}&−{5}&{8}\backslash{0}&{0}&{0}&{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$
The coefficient matrix for a system of linear differential equations of the form $$y^1=Ay$$
$$\lambda_1=2i \Rightarrow \left\{ \begin{bmatrix}1+i\\ 2-i \end{bmatrix} \right\} , \lambda_2=-2i \Rightarrow \left\{ \begin{bmatrix}1-i\\ 2+i \end{bmatrix} \right\}$$
Determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. All vectors (x, y, z) in $$V_3$$ whose components satisfy a system of three linear equations of the form:
$$a_{11}x+a_{12}y+a_{13}z=0$$
$$a_{21}x+a_{22}y+a_{23}z=0$$
$$a_{31}x+a_{32}y+a_{33}z=0$$