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

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

Solve for x, y, z

$$x=3$$

$$y=2$$

$$z=1$$

Short Steps Using Substitution

Solve 2x-3y+5z=5 for x.

$$x=\frac{3}{2}y-\frac{5}{2}z+\frac{5}{2}$$

Substitute \frac{3}{2}y-\frac{5}{2}z+\frac{5}{2} for x in the second and third equation.

$$4\left(\frac{3}{2}y-\frac{5}{2}z+\frac{5}{2}\right)-5y-7z=-5 -6\left(\frac{3}{2}y-\frac{5}{2}z+\frac{5}{2}\right)+7y+2z=-2$$

Solve these equations for y and z respectively.

$$y=-15+17z z=\frac{13}{17}+\frac{2}{17}y$$

Substitute -15+17z for y in the equation z=\frac{13}{17}+\frac{2}{17}y.

$$z=\frac{13}{17}+\frac{2}{17}\left(-15+17z\right)$$

Solve z=\frac{13}{17}+\frac{2}{17}\left(-15+17z\right) for z.

$$z=1$$

Substitute 1 for z in the equation y=-15+17z.

$$y=-15+17\times 1$$

Calculate y from y=-15+17\times 1.

$$y=2$$

Substitute 2 for y and 1 for z in the equation x=\frac{3}{2}y-\frac{5}{2}z+\frac{5}{2}.

$$x=\frac{3}{2}\times 2-\frac{5}{2}+\frac{5}{2}$$

Calculate x from x=\frac{3}{2}\times 2-\frac{5}{2}+\frac{5}{2}.

$$x=3$$

The system is now solved.

$$x=3 y=2 z=1$$