Question

Check whether the following system of equations has a solution
x + y + 3z + w = 5
-x + y + z - 5w = 7
x + 2y + 5z - w = 5

Answer 1

Answer on Question #68961 – Math – Linear Algebra

Question

Check whether the following system of equations has a solution

$$x + y + 3z + w = 5$$ $$x + y + z - 5w = 7$$ $$x + 2y + 5z - w = 5$$

Solution

Augmented matrix

$$\left[ \begin{array}{ccccc} 1 & 1 & 3 & 1 & 5 \\ -1 & 1 & 1 & -5 & 7 \\ 1 & 2 & 5 & -1 & 5 \end{array} \right]$$

Add row1 to row2 ($R_2 \leftarrow R_2 + R_1$)

$$\left[ \begin{array}{ccccc} 1 & 1 & 3 & 1 & 5 \\ 0 & 2 & 4 & -4 & 12 \\ 1 & 2 & 5 & -1 & 5 \end{array} \right]$$

Subtract row1 from row3 ($R_3 \leftarrow R_3 - R_1$)

$$\left[ \begin{array}{ccccc} 1 & 1 & 3 & 1 & 5 \\ 0 & 2 & 4 & -4 & 12 \\ 0 & 1 & 2 & -2 & 0 \end{array} \right]$$

Divide row2 by 2 ($R_2 \leftarrow R_2/2$)

$$\left[ \begin{array}{ccccc} 1 & 1 & 3 & 1 & 5 \\ 0 & 1 & 2 & -2 & 6 \\ 0 & 1 & 2 & -2 & 0 \end{array} \right]$$

Subtract row2 from row1 ($R_1 \leftarrow R_1 - R_2$)

$$\left[ \begin{array}{ccccc} 1 & 0 & 1 & 3 & -1 \\ 0 & 1 & 2 & -2 & 6 \\ 0 & 1 & 2 & -2 & 0 \end{array} \right]$$

Subtract row2 from row3 ($R_3 \leftarrow R_3 - R_2$)

$$\left[ \begin{array}{ccccc} 1 & 0 & 1 & 3 & -1 \\ 0 & 1 & 2 & -2 & 6 \\ 0 & 0 & 0 & 0 & -6 \end{array} \right]$$

Add row3 to row2 ($R_2 \leftarrow R_2 + R_3$)

$$\left[ \begin{array}{ccccc} 1 & 0 & 1 & 3 & -1 \\ 0 & 1 & 2 & -2 & 0 \\ 0 & 0 & 0 & 0 & -6 \end{array} \right]$$

Divide row3 by $-6$ ($R_3 \leftarrow R_3/(-6)$)

$$\left[ \begin{array}{ccccc} 1 & 0 & 1 & 3 & -1 \\ 0 & 1 & 2 & -2 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right]$$

Add row3 to row1 ($R_1 \leftarrow R_1 + R_3$)

$$\left[ \begin{array}{ccccc} 1 & 0 & 1 & 3 & 0 \\ 0 & 1 & 2 & -2 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right]$$

\left\{ \begin{array}{c} x + z - w = 0 \\ y + 2 z - 2 w = 0 \\ 0 = 1 \end{array} \right.

$$

The system is inconsistent and has no solution.

**Answer:** the system has no solution.

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