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

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

Solve for x, y, z

$$x=\frac{19}{21}\approx 0.904761905$$

$$y = \frac{43}{21} = 2\frac{1}{21} \approx 2.047619048$$

$$z = -\frac{32}{21} = -1\frac{11}{21} \approx -1.523809524$$

Short Steps Using Substitution

Solve x+2y=5 for x.

$$x=-2y+5$$

Substitute -2y+5 for x in the equation z+5x=3.

$$z+5\left(-2y+5\right)=3$$

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

$$y=-2z-1 z=-22+10y$$

Substitute -2z-1 for y in the equation z=-22+10y.

$$z=-22+10\left(-2z-1\right)$$

Solve z=-22+10\left(-2z-1\right) for z.

$$z=-\frac{32}{21}$$

Substitute -\frac{32}{21} for z in the equation y=-2z-1.

$$y=-2\left(-\frac{32}{21}\right)-1$$

Calculate y from y=-2\left(-\frac{32}{21}\right)-1.

$$y=\frac{43}{21}$$

Substitute \frac{43}{21} for y in the equation x=-2y+5.

$$x=-2\times \frac{43}{21}+5$$

Calculate x from x=-2\times \frac{43}{21}+5.

$$x=\frac{19}{21}$$

The system is now solved.

$$x=\frac{19}{21} y=\frac{43}{21} z=-\frac{32}{21}$$