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

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

Solve for x, y, z

$$x = \frac{301}{101} = 2\frac{99}{101} \approx 2.98019802$$

$$y=\frac{29}{101}\approx 0.287128713$$

$$z = \frac{316}{101} = 3\frac{13}{101} \approx 3.128712871$$

Short Steps Using Substitution

Reorder the equations.

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

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

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

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

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

Solve these equations for x and z respectively.

$$x=\frac{32}{11}+\frac{1}{44}z z=-\frac{83}{3}+\frac{31}{3}x$$

Substitute \frac{32}{11}+\frac{1}{44}z for x in the equation z=-\frac{83}{3}+\frac{31}{3}x.

$$z=-\frac{83}{3}+\frac{31}{3}\left(\frac{32}{11}+\frac{1}{44}z\right)$$

Solve z=-\frac{83}{3}+\frac{31}{3}\left(\frac{32}{11}+\frac{1}{44}z\right) for z.

$$z=\frac{316}{101}$$

Substitute \frac{316}{101} for z in the equation x=\frac{32}{11}+\frac{1}{44}z.

$$x=\frac{32}{11}+\frac{1}{44}\times \frac{316}{101}$$

Calculate x from x=\frac{32}{11}+\frac{1}{44}\times \frac{316}{101}.

$$x=\frac{301}{101}$$

Substitute \frac{301}{101} for x and \frac{316}{101} for z in the equation y=-8x+z+21.

$$y=-8\times \frac{301}{101}+\frac{316}{101}+21$$

Calculate y from y=-8\times \frac{301}{101}+\frac{316}{101}+21.

$$y=\frac{29}{101}$$

The system is now solved.

$$x=\frac{301}{101} y=\frac{29}{101} z=\frac{316}{101}$$