Solve {l}{x+y+z=4}{2x-5y=0}{y+3z=-5}

$$\left\{ \begin{array} { l } { x + y + z = 4 } \\ { 2 x - 5 y = 0 } \\ { y + 3 z = - 5 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{85}{19} = 4\frac{9}{19} \approx 4.473684211$$

$$y = \frac{34}{19} = 1\frac{15}{19} \approx 1.789473684$$

$$z = -\frac{43}{19} = -2\frac{5}{19} \approx -2.263157895$$

Short Steps Using Substitution

Solve x+y+z=4 for x.

$$x=-y-z+4$$

Substitute -y-z+4 for x in the equation 2x-5y=0.

$$2\left(-y-z+4\right)-5y=0$$

Solve the second equation for y and the third equation for z.

$$y=\frac{8}{7}-\frac{2}{7}z z=-\frac{1}{3}y-\frac{5}{3}$$

Substitute \frac{8}{7}-\frac{2}{7}z for y in the equation z=-\frac{1}{3}y-\frac{5}{3}.

$$z=-\frac{1}{3}\left(\frac{8}{7}-\frac{2}{7}z\right)-\frac{5}{3}$$

Solve z=-\frac{1}{3}\left(\frac{8}{7}-\frac{2}{7}z\right)-\frac{5}{3} for z.

$$z=-\frac{43}{19}$$

Substitute -\frac{43}{19} for z in the equation y=\frac{8}{7}-\frac{2}{7}z.

$$y=\frac{8}{7}-\frac{2}{7}\left(-\frac{43}{19}\right)$$

Calculate y from y=\frac{8}{7}-\frac{2}{7}\left(-\frac{43}{19}\right).

$$y=\frac{34}{19}$$

Substitute \frac{34}{19} for y and -\frac{43}{19} for z in the equation x=-y-z+4.

$$x=-\frac{34}{19}-\left(-\frac{43}{19}\right)+4$$

Calculate x from x=-\frac{34}{19}-\left(-\frac{43}{19}\right)+4.

$$x=\frac{85}{19}$$

The system is now solved.

$$x=\frac{85}{19} y=\frac{34}{19} z=-\frac{43}{19}$$