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

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

Solve for x, y, z

$$x=-\frac{7}{19}\approx -0.368421053$$

$$y=-\frac{17}{19}\approx -0.894736842$$

$$z=\frac{12}{19}\approx 0.631578947$$

Short Steps Using Substitution

Reorder the equations.

$$1+x=z 3x-4y+4z=5 -2x+5y-2z=-5$$

Solve 1+x=z for z.

$$z=1+x$$

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

$$3x-4y+4\left(1+x\right)=5 -2x+5y-2\left(1+x\right)=-5$$

Solve these equations for y and x respectively.

$$y=\frac{7}{4}x-\frac{1}{4} x=\frac{5}{4}y+\frac{3}{4}$$

Substitute \frac{7}{4}x-\frac{1}{4} for y in the equation x=\frac{5}{4}y+\frac{3}{4}.

$$x=\frac{5}{4}\left(\frac{7}{4}x-\frac{1}{4}\right)+\frac{3}{4}$$

Solve x=\frac{5}{4}\left(\frac{7}{4}x-\frac{1}{4}\right)+\frac{3}{4} for x.

$$x=-\frac{7}{19}$$

Substitute -\frac{7}{19} for x in the equation y=\frac{7}{4}x-\frac{1}{4}.

$$y=\frac{7}{4}\left(-\frac{7}{19}\right)-\frac{1}{4}$$

Calculate y from y=\frac{7}{4}\left(-\frac{7}{19}\right)-\frac{1}{4}.

$$y=-\frac{17}{19}$$

Substitute -\frac{7}{19} for x in the equation z=1+x.

$$z=1-\frac{7}{19}$$

Calculate z from z=1-\frac{7}{19}.

$$z=\frac{12}{19}$$

The system is now solved.

$$x=-\frac{7}{19} y=-\frac{17}{19} z=\frac{12}{19}$$