Solve {l}{3x-5y+3z=28}{5x-4y-2z=15}{x-y+4z=33}

$$\left. \begin{array}{l}{ 3 x - 5 y + 3 z = 28 }\\{ 5 x - 4 y - 2 z = 15 }\\{ x - y + 4 z = 33 }\end{array} \right.$$

Solve for x, y, z

$$x=9$$

$$y=4$$

$$z=7$$

Short Steps Using Substitution

Reorder the equations.

$$x-y+4z=33 5x-4y-2z=15 3x-5y+3z=28$$

Solve x-y+4z=33 for x.

$$x=y-4z+33$$

Substitute y-4z+33 for x in the second and third equation.

$$5\left(y-4z+33\right)-4y-2z=15 3\left(y-4z+33\right)-5y+3z=28$$

Solve these equations for y and z respectively.

$$y=-150+22z z=-\frac{2}{9}y+\frac{71}{9}$$

Substitute -150+22z for y in the equation z=-\frac{2}{9}y+\frac{71}{9}.

$$z=-\frac{2}{9}\left(-150+22z\right)+\frac{71}{9}$$

Solve z=-\frac{2}{9}\left(-150+22z\right)+\frac{71}{9} for z.

$$z=7$$

Substitute 7 for z in the equation y=-150+22z.

$$y=-150+22\times 7$$

Calculate y from y=-150+22\times 7.

$$y=4$$

Substitute 4 for y and 7 for z in the equation x=y-4z+33.

$$x=4-4\times 7+33$$

Calculate x from x=4-4\times 7+33.

$$x=9$$

The system is now solved.

$$x=9 y=4 z=7$$