Solve {l}{2x+4y-12z=6}{x+y+2z=26}{3x-3y+3z=30}

$$\left. \begin{array} { l } { 2 x + 4 y - 12 z = 6 } \\ { x + y + 2 z = 26 } \\ { 3 x - 3 y + 3 z = 30 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{213}{17} = 12\frac{9}{17} \approx 12.529411765$$

$$y = \frac{105}{17} = 6\frac{3}{17} \approx 6.176470588$$

$$z = \frac{62}{17} = 3\frac{11}{17} \approx 3.647058824$$

Short Steps Using Substitution

Reorder the equations.

$$x+y+2z=26 2x+4y-12z=6 3x-3y+3z=30$$

Solve x+y+2z=26 for x.

$$x=-y-2z+26$$

Substitute -y-2z+26 for x in the second and third equation.

$$2\left(-y-2z+26\right)+4y-12z=6 3\left(-y-2z+26\right)-3y+3z=30$$

Solve these equations for y and z respectively.

$$y=-23+8z z=16-2y$$

Substitute -23+8z for y in the equation z=16-2y.

$$z=16-2\left(-23+8z\right)$$

Solve z=16-2\left(-23+8z\right) for z.

$$z=\frac{62}{17}$$

Substitute \frac{62}{17} for z in the equation y=-23+8z.

$$y=-23+8\times \frac{62}{17}$$

Calculate y from y=-23+8\times \frac{62}{17}.

$$y=\frac{105}{17}$$

Substitute \frac{105}{17} for y and \frac{62}{17} for z in the equation x=-y-2z+26.

$$x=-\frac{105}{17}-2\times \frac{62}{17}+26$$

Calculate x from x=-\frac{105}{17}-2\times \frac{62}{17}+26.

$$x=\frac{213}{17}$$

The system is now solved.

$$x=\frac{213}{17} y=\frac{105}{17} z=\frac{62}{17}$$