# 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}

Question
Forms of linear equations
Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations. $$\begin{cases}8x+3y=25 \\ 3x-9y=12 \end{cases}$$

2020-11-11

For a system of equations $$\begin{cases}ax+by=j \\ cx+dy=k \end{cases}$$ ,the coefficient matrix is $$\begin{bmatrix} a&b \\ c&d \end{bmatrix}$$ and the augmented matrix is $$\begin{bmatrix}a&b&j\\ c&d&k \end{bmatrix}$$
a) For the system $$\begin{cases}8x+3y=25 \\ 3x-9y=12 \end{cases} a=8,b=3,c=3\ and\ d=-9$$ so the coefficient matrix is $$\begin{bmatrix}8&3\\ 3&-9 \end{bmatrix}$$
b) For the system $$\begin{cases}8x+3y=25 \\ 3x-9y=12 \end{cases} a=8,b=3,c=3 , d=-9 , j=25\ and\ k=12$$ so the augmented matrix is $$\begin{bmatrix}8&3&25\\ 3&-9&12 \end{bmatrix}$$

### Relevant Questions

Form (a) the coefficient matrix and (b) the augmented matrix for the system of linear equations $$\begin{cases}9x-3y+z=13 \\ 12x-8z=5 \\ 3x+4y-z =6 \end{cases}$$

Find the augmented matrix for the following system of linear equations:
$$\begin{cases}5x+7y-36z=38\\-8x-11y+57z=-60\end{cases}$$

Find the augmented matrix for the following system of linear equations:
$$3x+7y-20z=-4$$
$$5x+12y-34z=-7$$

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

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. $$\begin{bmatrix} 1 & 0 & −1 & 3 & 9\\ 0 & 1& 2 & −5 & 8\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$$

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.

Find the solution (x,y)to the system of equations.
$$\begin{cases}3x + y = 28\\-x+3y=4\end{cases}$$
Multiply the coordinates $$x \cdot y$$

The coefficient matrix for a system of linear differential equations of the form $$\displaystyle{y}^{{{1}}}={A}_{{{y}}}$$ has the given eigenvalues and eigenspace bases. Find the general solution for the system.
$$\left[\lambda_{1}=-1\Rightarrow\left\{\begin{bmatrix}1 0 3 \end{bmatrix}\right\},\lambda_{2}=3i\Rightarrow\left\{\begin{bmatrix}2-i 1+i 7i \end{bmatrix}\right\},\lambda_3=-3i\Rightarrow\left\{\begin{bmatrix}2+i 1-i -7i \end{bmatrix}\right\}\right]$$

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
$$\lambda1=3\Rightarrow \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$$
$$\lambda2=0\Rightarrow \begin{bmatrix} 1 \\ 5 \\ 1 \end{bmatrix}\begin{bmatrix}2 \\ 1 \\ 4 \end{bmatrix}$$
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.