Question

Check whether the following system of equations has a solution. 4x+2y+8z+6z=3 2x+2y+2z+2w=1 x+3z+2w=3?

Answer 1

Answer on Question #84623 – Math – Linear Algebra

Question

Check whether the following system of equations has a solution:

$$4x + 2y + 8z + 6z = 3, 2x + 2y + 2z + 2w = 1, x + 3z + 2w = 3?$$

Solution

Find $(x, y, z, w)$ for this system

$$\left\{ \begin{array}{l} x + 3z + 2w = 3, \\ 2x + 2y + 2z + 2w = 1, \\ 4x + 2y + 8z + 6z = 3. \end{array} \right.$$

Construct the following matrix and reducing it to a triangular form

$$\begin{array}{l} \left[ \begin{array}{cccc} 1 & 0 & 3 & 2 & 3 \\ 2 & 2 & 2 & 2 & 1 \\ 4 & 2 & 14 & 0 & 3 \end{array} \right] 2 \text{ row} + 1 \text{ row} \times (-2), 3 \text{ row} + 1 \text{ row} \times (-4) \\ \rightarrow \left[ \begin{array}{cccc} 1 & 0 & 3 & 2 & 3 \\ 0 & 2 & -4 & -2 & -5 \\ 0 & 2 & 2 & -8 & -9 \end{array} \right] 3 \text{ row} + 2 \text{ row} \times (-1) \rightarrow \left[ \begin{array}{cccc} 1 & 0 & 3 & 2 & 3 \\ 0 & 2 & -4 & -2 & -5 \\ 0 & 0 & 6 & -6 & -4 \end{array} \right]. \end{array}$$

The system that corresponds to the last matrix has the form

$$\left.\begin{array}{l} x + 3z + 2w = 3 \\ 2y - 4z - 2w = -5 \\ 6z - 6w = -4. \end{array} \right\} \Rightarrow \left.\begin{array}{l} x = 3 - 3z - 2w \\ y = \dfrac{-5 + 4z + 2w}{2} \\ z = \dfrac{3w - 2}{3} \\ w = w \end{array} \right\} \Rightarrow \left.\begin{array}{l} x = 5 - 5w \\ y = \dfrac{18w - 23}{6} \\ z = \dfrac{3w - 2}{3} \\ w = w \end{array} \right\}$$

Answer: yes, $x = 5 - 5w$, $y = \dfrac{18w - 23}{6}$, $z = \dfrac{3w - 2}{3}$, $w = w$, where $w \in \mathbb{R}$.

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