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

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

Solve for x, y, z

$$x=7$$

$$y=3$$

$$z=1$$

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.

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

Solve these equations for x and z respectively.

$$x=\frac{105}{11}-\frac{28}{11}z z=-\frac{11}{19}x+\frac{96}{19}$$

Substitute \frac{105}{11}-\frac{28}{11}z for x in the equation z=-\frac{11}{19}x+\frac{96}{19}.

$$z=-\frac{11}{19}\left(\frac{105}{11}-\frac{28}{11}z\right)+\frac{96}{19}$$

Solve z=-\frac{11}{19}\left(\frac{105}{11}-\frac{28}{11}z\right)+\frac{96}{19} for z.

$$z=1$$

Substitute 1 for z in the equation x=\frac{105}{11}-\frac{28}{11}z.

$$x=\frac{105}{11}-\frac{28}{11}$$

Calculate x from x=\frac{105}{11}-\frac{28}{11}.

$$x=7$$

Substitute 7 for x and 1 for z in the equation y=2x+5z-16.

$$y=2\times 7+5\times 1-16$$

Calculate y from y=2\times 7+5\times 1-16.

$$y=3$$

The system is now solved.

$$x=7 y=3 z=1$$