Under which conditions is the following linear equation system solvable $x_1 + 2x_2 - 3x_3 = a$ $3x_1 - x_2 + 2x_3 = b$ $x_1 - 5x_2 + 8x_3 = c$

Question

I'm unsure about the problem below

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Under which conditions is the following linear equation system solvable ? $$x_1 + 2x_2 - 3x_3 = a$$ $$3x_1 - x_2 + 2x_3 = b$$ $$x_1 - 5x_2 + 8x_3 = c$$

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We set up our matrix

$$\begin{bmatrix} 1 & 2 & -3 & | a \\ 3 & -1 & 2 & | b \\ 1 & -5 & 8 & | c \\ \end{bmatrix}$$

We apply -3 first row to second row and -1 first row to third row. Then we add -1 second row to third row. We get

$$\begin{bmatrix} 1 & 2 & -3 & |a\\ 0 & -7 & 11 & |b - 3a\\ 0 & 0 & 0 & |2a - b + c\\ \end{bmatrix}$$

So $2a - b + c = 0$ for the system to be solvable. Is this correct ? I fear that there are other conditions that I forgot ?

Answer 1

Yes, you are correct. The point is you have a (bottom) row with all zeros, and on that row, the right side must be zero. You are in row echelon form if you take full course on linear algebra, and often the first chapter of those books will address question like this.

Answer 2

A linear system is inconsistent (i.e. it has no solution) if and only if the augmented matrix has a pivot in the last column. Since you've put the matrix in row echelon form, we can clearly identify the pivot positions. When $2a-b+c=0$, there are 2 pivots and none of them are in the last column. Thus, the system will have a solution.