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

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

Solve for x, y, z

$$x=1$$

$$y=-1$$

$$z=-2$$

Short Steps Using Substitution

Reorder the equations.

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

Solve 2x+y-z=3 for y.

$$y=-2x+z+3$$

Substitute -2x+z+3 for y in the second and third equation.

$$3x-5\left(-2x+z+3\right)=8 3x-5\left(-2x+z+3\right)+2z=4$$

Solve these equations for x and z respectively.

$$x=\frac{5}{13}z+\frac{23}{13} z=\frac{13}{3}x-\frac{19}{3}$$

Substitute \frac{5}{13}z+\frac{23}{13} for x in the equation z=\frac{13}{3}x-\frac{19}{3}.

$$z=\frac{13}{3}\left(\frac{5}{13}z+\frac{23}{13}\right)-\frac{19}{3}$$

Solve z=\frac{13}{3}\left(\frac{5}{13}z+\frac{23}{13}\right)-\frac{19}{3} for z.

$$z=-2$$

Substitute -2 for z in the equation x=\frac{5}{13}z+\frac{23}{13}.

$$x=\frac{5}{13}\left(-2\right)+\frac{23}{13}$$

Calculate x from x=\frac{5}{13}\left(-2\right)+\frac{23}{13}.

$$x=1$$

Substitute 1 for x and -2 for z in the equation y=-2x+z+3.

$$y=-2-2+3$$

Calculate y from y=-2-2+3.

$$y=-1$$

The system is now solved.

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