Solve {l}{2x+y+2=1}{2x-4y-2z=3}{3y-5z=9}

$$\left. \begin{array} { l } { 2 x + y + 2 = 1 } \\ { 2 x - 4 y - 2 z = 3 } \\ { 3 y - 5 z = 9 } \end{array} \right.$$

Solve for x, y, z

$$x=-\frac{29}{62}\approx -0.467741935$$

$$y=-\frac{2}{31}\approx -0.064516129$$

$$z = -\frac{57}{31} = -1\frac{26}{31} \approx -1.838709677$$

Short Steps Using Substitution

Solve 2x+y+2=1 for y.

$$y=-2x-1$$

Substitute -2x-1 for y in the second and third equation.

$$2x-4\left(-2x-1\right)-2z=3 3\left(-2x-1\right)-5z=9$$

Solve these equations for x and z respectively.

$$x=-\frac{1}{10}+\frac{1}{5}z z=-\frac{12}{5}-\frac{6}{5}x$$

Substitute -\frac{1}{10}+\frac{1}{5}z for x in the equation z=-\frac{12}{5}-\frac{6}{5}x.

$$z=-\frac{12}{5}-\frac{6}{5}\left(-\frac{1}{10}+\frac{1}{5}z\right)$$

Solve z=-\frac{12}{5}-\frac{6}{5}\left(-\frac{1}{10}+\frac{1}{5}z\right) for z.

$$z=-\frac{57}{31}$$

Substitute -\frac{57}{31} for z in the equation x=-\frac{1}{10}+\frac{1}{5}z.

$$x=-\frac{1}{10}+\frac{1}{5}\left(-\frac{57}{31}\right)$$

Calculate x from x=-\frac{1}{10}+\frac{1}{5}\left(-\frac{57}{31}\right).

$$x=-\frac{29}{62}$$

Substitute -\frac{29}{62} for x in the equation y=-2x-1.

$$y=-2\left(-\frac{29}{62}\right)-1$$

Calculate y from y=-2\left(-\frac{29}{62}\right)-1.

$$y=-\frac{2}{31}$$

The system is now solved.

$$x=-\frac{29}{62} y=-\frac{2}{31} z=-\frac{57}{31}$$