Solve {l}{x+2y+z=1}{7x+3y-z=-2}{x+5y+3z=2}

$$\left. \begin{array} { l } { x + 2 y + z = 1 } \\ { 7 x + 3 y - z = - 2 } \\ { x + 5 y + 3 z = 2 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=-5$$

$$z=8$$

Short Steps Using Substitution

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

$$x=-2y-z+1$$

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

$$7\left(-2y-z+1\right)+3y-z=-2 -2y-z+1+5y+3z=2$$

Solve these equations for y and z respectively.

$$y=-\frac{8}{11}z+\frac{9}{11} z=-\frac{3}{2}y+\frac{1}{2}$$

Substitute -\frac{8}{11}z+\frac{9}{11} for y in the equation z=-\frac{3}{2}y+\frac{1}{2}.

$$z=-\frac{3}{2}\left(-\frac{8}{11}z+\frac{9}{11}\right)+\frac{1}{2}$$

Solve z=-\frac{3}{2}\left(-\frac{8}{11}z+\frac{9}{11}\right)+\frac{1}{2} for z.

$$z=8$$

Substitute 8 for z in the equation y=-\frac{8}{11}z+\frac{9}{11}.

$$y=-\frac{8}{11}\times 8+\frac{9}{11}$$

Calculate y from y=-\frac{8}{11}\times 8+\frac{9}{11}.

$$y=-5$$

Substitute -5 for y and 8 for z in the equation x=-2y-z+1.

$$x=-2\left(-5\right)-8+1$$

Calculate x from x=-2\left(-5\right)-8+1.

$$x=3$$

The system is now solved.

$$x=3 y=-5 z=8$$