Solve {l}{-2x+3y-5z=-9}{x+y-2z=3}{5x+4y+z=34}

$$\left. \begin{array}{l}{ - 2 x + 3 y - 5 z = - 9 }\\{ x + y - 2 z = 3 }\\{ 5 x + 4 y + z = 34 }\end{array} \right.$$

Solve for x, y, z

$$x=4$$

$$y=3$$

$$z=2$$

Short Steps Using Substitution

Reorder the equations.

$$x+y-2z=3 -2x+3y-5z=-9 5x+4y+z=34$$

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

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

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

$$-2\left(-y+2z+3\right)+3y-5z=-9 5\left(-y+2z+3\right)+4y+z=34$$

Solve these equations for y and z respectively.

$$y=-\frac{3}{5}+\frac{9}{5}z z=\frac{19}{11}+\frac{1}{11}y$$

Substitute -\frac{3}{5}+\frac{9}{5}z for y in the equation z=\frac{19}{11}+\frac{1}{11}y.

$$z=\frac{19}{11}+\frac{1}{11}\left(-\frac{3}{5}+\frac{9}{5}z\right)$$

Solve z=\frac{19}{11}+\frac{1}{11}\left(-\frac{3}{5}+\frac{9}{5}z\right) for z.

$$z=2$$

Substitute 2 for z in the equation y=-\frac{3}{5}+\frac{9}{5}z.

$$y=-\frac{3}{5}+\frac{9}{5}\times 2$$

Calculate y from y=-\frac{3}{5}+\frac{9}{5}\times 2.

$$y=3$$

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

$$x=-3+2\times 2+3$$

Calculate x from x=-3+2\times 2+3.

$$x=4$$

The system is now solved.

$$x=4 y=3 z=2$$