Solve {l}{2x+y-3z=5}{-4x-y+2z=12}{5x+y+7z=14}

$$\left\{ \begin{array} { l } { 2 x + y - 3 z = 5 } \\ { - 4 x - y + 2 z = 12 } \\ { 5 x + y + 7 z = 14 } \end{array} \right.$$

Solve for x, y, z

$$x = -\frac{179}{17} = -10\frac{9}{17} \approx -10.529411765$$

$$y = \frac{650}{17} = 38\frac{4}{17} \approx 38.235294118$$

$$z = \frac{69}{17} = 4\frac{1}{17} \approx 4.058823529$$

Short Steps Using Substitution

Solve 2x+y-3z=5 for y.

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

Substitute -2x+3z+5 for y in the second and third equation.

$$-4x-\left(-2x+3z+5\right)+2z=12 5x-2x+3z+5+7z=14$$

Solve these equations for x and z respectively.

$$x=-\frac{1}{2}z-\frac{17}{2} z=\frac{9}{10}-\frac{3}{10}x$$

Substitute -\frac{1}{2}z-\frac{17}{2} for x in the equation z=\frac{9}{10}-\frac{3}{10}x.

$$z=\frac{9}{10}-\frac{3}{10}\left(-\frac{1}{2}z-\frac{17}{2}\right)$$

Solve z=\frac{9}{10}-\frac{3}{10}\left(-\frac{1}{2}z-\frac{17}{2}\right) for z.

$$z=\frac{69}{17}$$

Substitute \frac{69}{17} for z in the equation x=-\frac{1}{2}z-\frac{17}{2}.

$$x=-\frac{1}{2}\times \frac{69}{17}-\frac{17}{2}$$

Calculate x from x=-\frac{1}{2}\times \frac{69}{17}-\frac{17}{2}.

$$x=-\frac{179}{17}$$

Substitute -\frac{179}{17} for x and \frac{69}{17} for z in the equation y=-2x+3z+5.

$$y=-2\left(-\frac{179}{17}\right)+3\times \frac{69}{17}+5$$

Calculate y from y=-2\left(-\frac{179}{17}\right)+3\times \frac{69}{17}+5.

$$y=\frac{650}{17}$$

The system is now solved.

$$x=-\frac{179}{17} y=\frac{650}{17} z=\frac{69}{17}$$