Solve {r}{2x-y+5z=16}{2-6y+2z=-9}{3x+4y-z=32}

$$\left\{ \begin{array} { r } { 2 x - y + 5 z = 16 } \\ { 2 - 6 y + 2 z = - 9 } \\ { 3 x + 4 y - z = 32 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{131}{16} = 8\frac{3}{16} = 8.1875$$

$$y = \frac{31}{16} = 1\frac{15}{16} = 1.9375$$

$$z=\frac{5}{16}=0.3125$$

Short Steps Using Substitution

Solve 2x-y+5z=16 for y.

$$y=2x+5z-16$$

Substitute 2x+5z-16 for y in the second and third equation.

$$2-6\left(2x+5z-16\right)+2z=-9 3x+4\left(2x+5z-16\right)-z=32$$

Solve these equations for x and z respectively.

$$x=\frac{107}{12}-\frac{7}{3}z z=-\frac{11}{19}x+\frac{96}{19}$$

Substitute \frac{107}{12}-\frac{7}{3}z for x in the equation z=-\frac{11}{19}x+\frac{96}{19}.

$$z=-\frac{11}{19}\left(\frac{107}{12}-\frac{7}{3}z\right)+\frac{96}{19}$$

Solve z=-\frac{11}{19}\left(\frac{107}{12}-\frac{7}{3}z\right)+\frac{96}{19} for z.

$$z=\frac{5}{16}$$

Substitute \frac{5}{16} for z in the equation x=\frac{107}{12}-\frac{7}{3}z.

$$x=\frac{107}{12}-\frac{7}{3}\times \frac{5}{16}$$

Calculate x from x=\frac{107}{12}-\frac{7}{3}\times \frac{5}{16}.

$$x=\frac{131}{16}$$

Substitute \frac{131}{16} for x and \frac{5}{16} for z in the equation y=2x+5z-16.

$$y=2\times \frac{131}{16}+5\times \frac{5}{16}-16$$

Calculate y from y=2\times \frac{131}{16}+5\times \frac{5}{16}-16.

$$y=\frac{31}{16}$$

The system is now solved.

$$x=\frac{131}{16} y=\frac{31}{16} z=\frac{5}{16}$$