# 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 & 2 & -1 0 & 1 & -4 & -20&0&0&0&0 end{matrix}

Question
Matrices
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.
$$\begin{matrix}1 & 0 & 2 & -1 \\ 0 & 1 & -4 & -2\\0&0&0&0&0 \end{matrix}$$

2021-02-01
a. The system represented by the matrix is
$$\begin{cases}x+2z=-1\\y-4z=-2\\0=0\end{cases}$$
b. This is a consistent system (because the last equation is true for all ordered triplets).
The system will not have a single solution, as it is, in effect, a system of TWO equations in THREE variables.
To solve, we take z to be any real number $$t\in\mathbb{R}$$, and back-substitute into the other two equations:
$$\begin{cases}x+2(t)=-1\\x=-1-2t\end{cases}$$
$$\begin{cases}y-4(t)=-2\\y=-2+4t\end{cases}$$
Result:
a. $$\begin{cases}x+2z=-1\\y-4z=-2\\0=0\end{cases}$$
b. Consistent, solution set: $$\left\{(-1+2t, -2+4t,t),\ t\in\mathbb{R}\right\}$$

### Relevant Questions

Use x, y, or x, y, z, or $$x_1,x_2,x_3, x_4$$
$$\begin{matrix}1 & 0 & 3 & 0 &1 \\ 0 & 1 & 4 & 3&2\\0&0&1&2&3\\0&0&0&0&0 \end{matrix}$$
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 2020-12-03 The following matrix is the augmented matrix of a system of linear equations in the variables x, y, and z. (It is given in reduced row-echelon form.) $$\begin{bmatrix}1&0&-1&3\\0&1&2&5\\0&0&0&0\end{bmatrix}$$ Find: (a) The leading variables, (b) Is the system in consistent or dependent? (c) The solution of the system. asked 2021-01-16 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. $$\displaystyle{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}&−{1}&{3}&{9}\backslash{0}&{1}&{2}&−{5}&{8}\backslash{0}&{0}&{0}&{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}$$ asked 2020-12-24 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. $$\begin{bmatrix}1&-2&0&0&-3\\0&0&1&0&-4\\0&0&0&1&5\end{bmatrix}$$ asked 2020-11-12 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. $$\begin{bmatrix}1&3&0&-2&6\\0&0&1&4&7\\0&0&0&0&0\end{bmatrix}$$ 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 2021-02-24 Let $$u=\begin{bmatrix}2 \\ 5 \\ -1 \end{bmatrix} , v=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix} \text{ and } w=\begin{bmatrix}-4 \\ 17 \\ -13 \end{bmatrix}$$ It can be shown that 4u-3v-w=0. Use this fact (and no row operations) to find a solution to the system Ax=b , where $$A=\begin{bmatrix}2 & -4 \\5 & 17\\-1&-13 \end{bmatrix} , x=\begin{bmatrix}x_1 \\ x_2 \end{bmatrix} , b=\begin{bmatrix}4 \\ 1 \\ 3 \end{bmatrix}$$ asked 2021-01-16 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 $$c_1$$. $$c_1\cdot\begin{cases}x_1+x_2+x_3=0\\5x_1-2x_2+2x_3=0\\8x_1+x_2+5x_3=0\end{cases},\ X=c_1\begin{pmatrix}4\\3\\-7\end{pmatrix}$$ 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.