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}

Question
Forms of linear equations
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. \(\displaystyle{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}&{0}&−{1}&{3}&{9}\backslash{0}&{1}&{2}&−{5}&{8}\backslash{0}&{0}&{0}&{0}&{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\)

Answers (1)

2021-01-17
\(\displaystyle{x}_{{{1}}}-{s}_{{{3}}}+{3}{x}_{{{4}}}={9}\)
\(\displaystyle{x}_{{{2}}}+{2}{x}_{{{3}}}-{5}{x}_{{{4}}}={8}\)
\(\displaystyle{0}={0}\) The augmented matrix does not contain a row in which the only nonzero entry appears in the last column. Therefore, this system of equations must be consistent. Convert the augmented matrix into a system of equations. \(\displaystyle{x}_{{1}}={9}+{x}_{{3}}-{3}{x}_{{4}}\)
\(\displaystyle{x}_{{2}}={8}-{2}{x}_{{3}}+{5}{x}_{{4}}\)
\(\displaystyle{x}_{{3}},{\mathfrak{{e}}}{e}\)
\(\displaystyle{x}_{{4}},{\mathfrak{{e}}}{e}\) Solve for the leading entry for each individual equation. Determine the free variables, if any. \(\displaystyle{x}_{{1}}={9}+{s}-{3}{t}\)
\(\displaystyle{x}_{{2}}={8}-{2}{s}+{5}{t}\)
\(\displaystyle{x}_{{3}}={s}\)
\(\displaystyle{x}_{{4}}={t}\) Parameterize the free variables. \(\displaystyle{x}={b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{9}\backslash{8}\backslash{0}\backslash{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}+{s}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}{1}\backslash-{2}\backslash{1}\backslash{0}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}+{t}{b}{e}{g}\in{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}-{3}\backslash{5}\backslash{0}\backslash{1}{e}{n}{d}{\left\lbrace{b}{m}{a}{t}{r}{i}{x}\right\rbrace}\) And write the solution in vector form.
0

Relevant Questions

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 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 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 2021-01-31
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.
\(\begin{matrix}1 & 0 & 2 & -1 \\ 0 & 1 & -4 & -2\\0&0&0&0&0 \end{matrix}\)
asked 2020-11-08
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
\(\displaystyle{\left[\begin{matrix}{1}&{0}&-{1}&-{2}\\{0}&{1}&{2}&{3}\end{matrix}\right]}\)
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 2020-11-01
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
\(\displaystyle{\left[\begin{matrix}{1}&-{1}&{0}&-{2}&{0}&{0}\\{0}&{0}&{1}&{2}&{0}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}\end{matrix}\right]}\)
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.
asked 2021-02-08
The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution. \(\begin{bmatrix}1&0&-1&-2&0\\0&1&2&3&0\end{bmatrix}\)
...