Solve {l}{6x+y-3z=5}{x+3y-2z=5}{2x+y+4z=8}

$$\left. \begin{array} { l } { 6 x + y - 3 z = 5 } \\ { x + 3 y - 2 z = 5 } \\ { 2 x + y + 4 z = 8 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y=2$$

$$z=1$$

Short Steps Using Substitution

Solve 6x+y-3z=5 for y.

$$y=-6x+3z+5$$

Substitute -6x+3z+5 for y in the second and third equation.

$$x+3\left(-6x+3z+5\right)-2z=5 2x-6x+3z+5+4z=8$$

Solve these equations for x and z respectively.

$$x=\frac{7}{17}z+\frac{10}{17} z=\frac{4}{7}x+\frac{3}{7}$$

Substitute \frac{7}{17}z+\frac{10}{17} for x in the equation z=\frac{4}{7}x+\frac{3}{7}.

$$z=\frac{4}{7}\left(\frac{7}{17}z+\frac{10}{17}\right)+\frac{3}{7}$$

Solve z=\frac{4}{7}\left(\frac{7}{17}z+\frac{10}{17}\right)+\frac{3}{7} for z.

$$z=1$$

Substitute 1 for z in the equation x=\frac{7}{17}z+\frac{10}{17}.

$$x=\frac{7}{17}\times 1+\frac{10}{17}$$

Calculate x from x=\frac{7}{17}\times 1+\frac{10}{17}.

$$x=1$$

Substitute 1 for x and 1 for z in the equation y=-6x+3z+5.

$$y=-6+3\times 1+5$$

Calculate y from y=-6+3\times 1+5.

$$y=2$$

The system is now solved.

$$x=1 y=2 z=1$$