Solve {l}{2x-5y+4z=6}{-8x+10y+5z=7}{x+y+z=3}

$$\left. \begin{array} { l } { 2 x - 5 y + 4 z = 6 } \\ { - 8 x + 10 y + 5 z = 7 } \\ { x + y + z = 3 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{102}{127}\approx 0.803149606$$

$$y=\frac{62}{127}\approx 0.488188976$$

$$z = \frac{217}{127} = 1\frac{90}{127} \approx 1.708661417$$

Short Steps Using Substitution

Reorder the equations.

$$x+y+z=3 -8x+10y+5z=7 2x-5y+4z=6$$

Solve x+y+z=3 for x.

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

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

$$-8\left(-y-z+3\right)+10y+5z=7 2\left(-y-z+3\right)-5y+4z=6$$

Solve these equations for y and z respectively.

$$y=-\frac{13}{18}z+\frac{31}{18} z=\frac{7}{2}y$$

Substitute -\frac{13}{18}z+\frac{31}{18} for y in the equation z=\frac{7}{2}y.

$$z=\frac{7}{2}\left(-\frac{13}{18}z+\frac{31}{18}\right)$$

Solve z=\frac{7}{2}\left(-\frac{13}{18}z+\frac{31}{18}\right) for z.

$$z=\frac{217}{127}$$

Substitute \frac{217}{127} for z in the equation y=-\frac{13}{18}z+\frac{31}{18}.

$$y=-\frac{13}{18}\times \frac{217}{127}+\frac{31}{18}$$

Calculate y from y=-\frac{13}{18}\times \frac{217}{127}+\frac{31}{18}.

$$y=\frac{62}{127}$$

Substitute \frac{62}{127} for y and \frac{217}{127} for z in the equation x=-y-z+3.

$$x=-\frac{62}{127}-\frac{217}{127}+3$$

Calculate x from x=-\frac{62}{127}-\frac{217}{127}+3.

$$x=\frac{102}{127}$$

The system is now solved.

$$x=\frac{102}{127} y=\frac{62}{127} z=\frac{217}{127}$$