Solve {l}{x+y+z=1}{2x+3y-2z=2}{6x+109y-100z=1}

$$\left\{ \begin{array} { l } { x + y + z = 1 } \\ { 2 x + 3 y - 2 z = 2 } \\ { 6 x + 109 y - 100 z = 1 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{331}{306} = 1\frac{25}{306} \approx 1.081699346$$

$$y=-\frac{10}{153}\approx -0.065359477$$

$$z=-\frac{5}{306}\approx -0.016339869$$

Short Steps Using Substitution

Solve x+y+z=1 for x.

$$x=-y-z+1$$

Substitute -y-z+1 for x in the second and third equation.

$$2\left(-y-z+1\right)+3y-2z=2 6\left(-y-z+1\right)+109y-100z=1$$

Solve these equations for y and z respectively.

$$y=4z z=\frac{103}{106}y+\frac{5}{106}$$

Substitute 4z for y in the equation z=\frac{103}{106}y+\frac{5}{106}.

$$z=\frac{103}{106}\times 4z+\frac{5}{106}$$

Solve z=\frac{103}{106}\times 4z+\frac{5}{106} for z.

$$z=-\frac{5}{306}$$

Substitute -\frac{5}{306} for z in the equation y=4z.

$$y=4\left(-\frac{5}{306}\right)$$

Calculate y from y=4\left(-\frac{5}{306}\right).

$$y=-\frac{10}{153}$$

Substitute -\frac{10}{153} for y and -\frac{5}{306} for z in the equation x=-y-z+1.

$$x=-\left(-\frac{10}{153}\right)-\left(-\frac{5}{306}\right)+1$$

Calculate x from x=-\left(-\frac{10}{153}\right)-\left(-\frac{5}{306}\right)+1.

$$x=\frac{331}{306}$$

The system is now solved.

$$x=\frac{331}{306} y=-\frac{10}{153} z=-\frac{5}{306}$$