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
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.
\(\begin{matrix}1 & 0 & 2 & -1 \\ 0 & 1 & -4 & -2\\0&0&0&0&0 \end{matrix}\)

Answers (1)

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: \(\left\{(-1+2t, -2+4t,t),\ t\in\mathbb{R}\right\}\)
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