Determine whether the system is consistent by setting up the appropriate augmented matrix and...

Question:

Determine whether the system is consistent by setting up the appropriate augmented matrix and using elementary row operations to find an echelon form of the matrix.

{eq}x_1 - x_2 + 3x_3 = -9 {/eq}

{eq}-5x_1 + 5x_2 - 15x_3 = -2 {/eq}

{eq}x_1 + 5x_2 + x_3 = -17{/eq}

Consistent Linear System

When we have a system of equations that relate multiple variables, this system doesn't always have a solution. If this system has at least one solution, then this system of equation is referred to as consistent. If it has no solutions, then it is inconsistent.

Answer and Explanation:

To find whether or not this system is consistent, we need to try to solve it. Let's construct an augmented matrix and perform row operations in the goal of putting this into reduced row echelon form. If we come across a contradiction at any time, we can stop and state that the system is inconsistent.

{eq}\begin{align*} \begin{bmatrix} 1 & -1 & 3 & -9\\ -5 & 5 & -15 & -2\\ 1 & 5 & 1 & -17 \end{bmatrix} && R_2 = R_2 + 5R_1\\ \begin{bmatrix} 1 & -1 & 3 & -9\\ 0 & 0 & 0 & -47\\ 1 & 5 & 1 & -17 \end{bmatrix} \end{align*} {/eq}

We can actually stop after this single calculation, since we've come across a contradiction. The middle row of this matrix states that {eq}0x_1 + 0x_2 + 0x_3 = -47 {/eq}, which is equivalent to the false statement {eq}0 = -47 {/eq}. This means that there are no solutions to this system, and that it is inconsistent.