Solve {l}{5x-3y+2z=20}{4x+6y-5z=-2}{8x+y-z=21}

$$\left. \begin{array} { l } { 5 x - 3 y + 2 z = 20 } \\ { 4 x + 6 y - 5 z = - 2 } \\ { 8 x + y - z = 21 } \end{array} \right.$$

Solve for x, y, z

$$x=3$$

$$y=1$$

$$z=4$$

Short Steps Using Substitution

Reorder the equations.

$$8x+y-z=21 4x+6y-5z=-2 5x-3y+2z=20$$

Solve 8x+y-z=21 for y.

$$y=-8x+z+21$$

Substitute -8x+z+21 for y in the second and third equation.

$$4x+6\left(-8x+z+21\right)-5z=-2 5x-3\left(-8x+z+21\right)+2z=20$$

Solve these equations for x and z respectively.

$$x=\frac{32}{11}+\frac{1}{44}z z=29x-83$$

Substitute \frac{32}{11}+\frac{1}{44}z for x in the equation z=29x-83.

$$z=29\left(\frac{32}{11}+\frac{1}{44}z\right)-83$$

Solve z=29\left(\frac{32}{11}+\frac{1}{44}z\right)-83 for z.

$$z=4$$

Substitute 4 for z in the equation x=\frac{32}{11}+\frac{1}{44}z.

$$x=\frac{32}{11}+\frac{1}{44}\times 4$$

Calculate x from x=\frac{32}{11}+\frac{1}{44}\times 4.

$$x=3$$

Substitute 3 for x and 4 for z in the equation y=-8x+z+21.

$$y=-8\times 3+4+21$$

Calculate y from y=-8\times 3+4+21.

$$y=1$$

The system is now solved.

$$x=3 y=1 z=4$$