QuestionUse Gauss elimination $$\left\{\begin{array}{l} x+2 y-z=-3 \\ 3 x-y+2 z=8 \\ 2 x+y-3 z=-5 \end{array}\right.$$

Studdy Solution
Perform back substitution to find the values of the variables.
From the third row, we have:
$$z = \frac{43}{8}$$
Substitute $z = \frac{43}{8}$ into the second row:
$$-7y + 5\left(\frac{43}{8}\right) = 17$$
Solve for $y$:
$$-7y + \frac{215}{8} = 17$$
$$-7y = 17 - \frac{215}{8}$$
$$-7y = \frac{136}{8} - \frac{215}{8}$$
$$-7y = -\frac{79}{8}$$
$$y = \frac{79}{56}$$
Substitute $y = \frac{79}{56}$ and $z = \frac{43}{8}$ into the first row:
$$x + 2\left(\frac{79}{56}\right) - \frac{43}{8} = -3$$
Solve for $x$:
$$x + \frac{158}{56} - \frac{301}{56} = -3$$
$$x - \frac{143}{56} = -3$$
$$x = -3 + \frac{143}{56}$$
$$x = -\frac{168}{56} + \frac{143}{56}$$
$$x = -\frac{25}{56}$$
The solution to the system of equations is:
$$x = -\frac{25}{56}, \, y = \frac{79}{56}, \, z = \frac{43}{8}$$