Solve {c}{4x-y+3z-6=0}{x+5y-z+10=0}{x=1}

$$\left. \begin{array} { c } { 4 x - y + 3 z - 6 = 0 } \\ { x + 5 y - z + 10 = 0 } \\ { x = 1 } \end{array} \right.$$

Solve for x, y, z

$$x=1$$

$$y = -\frac{31}{14} = -2\frac{3}{14} \approx -2.214285714$$

$$z=-\frac{1}{14}\approx -0.071428571$$

Short Steps Using Substitution

Reorder the equations.

$$x=1 x+5y-z+10=0 4x-y+3z-6=0$$

Substitute 1 for x in the second and third equation.

$$1+5y-z+10=0 4\times 1-y+3z-6=0$$

Solve these equations for y and z respectively.

$$y=-\frac{11}{5}+\frac{1}{5}z z=\frac{2}{3}+\frac{1}{3}y$$

Substitute -\frac{11}{5}+\frac{1}{5}z for y in the equation z=\frac{2}{3}+\frac{1}{3}y.

$$z=\frac{2}{3}+\frac{1}{3}\left(-\frac{11}{5}+\frac{1}{5}z\right)$$

Solve z=\frac{2}{3}+\frac{1}{3}\left(-\frac{11}{5}+\frac{1}{5}z\right) for z.

$$z=-\frac{1}{14}$$

Substitute -\frac{1}{14} for z in the equation y=-\frac{11}{5}+\frac{1}{5}z.

$$y=-\frac{11}{5}+\frac{1}{5}\left(-\frac{1}{14}\right)$$

Calculate y from y=-\frac{11}{5}+\frac{1}{5}\left(-\frac{1}{14}\right).

$$y=-\frac{31}{14}$$

The system is now solved.

$$x=1 y=-\frac{31}{14} z=-\frac{1}{14}$$