Questions: The reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding to the given matrix. Use x, y, and z as variables. Determine whether the system is consistent or inconsistent. If it is consistent, give the solution. [ [1 0 0 -1] [0 1 0 -3] [0 0 0 -6] ] What equation does the first row represent? (Type an equation.)

Solution Steps

The given matrix in reduced row echelon form is:

$$\begin{bmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 0 & -6 \end{bmatrix}$$

Each row of the matrix represents a linear equation. The first three columns correspond to the variables $x$, $y$, and $z$, and the last column represents the constants on the right-hand side of the equations.

• The first row, $[1, 0, 0, -1]$, represents the equation: $$1x + 0y + 0z = -1 \quad \Rightarrow \quad x = -1$$

• The second row, $[0, 1, 0, -3]$, represents the equation: $$0x + 1y + 0z = -3 \quad \Rightarrow \quad y = -3$$

• The third row, $[0, 0, 0, -6]$, represents the equation: $$0x + 0y + 0z = -6$$

A system of equations is consistent if there are no contradictions. A contradiction occurs when a row in the matrix has all zero coefficients for the variables but a non-zero constant on the right-hand side.

In this case, the third row is: $$0x + 0y + 0z = -6$$

This is a contradiction because it implies $0 = -6$, which is impossible. Therefore, the system is inconsistent.

Final Answer