Solve {l}{x-3y+2z=-3}{5x-46y-z=13}{4x-y+3z=8}

$$\left. \begin{array} { l } { x - 3 y + 2 z = - 3 } \\ { 5 x - 46 y - z = 13 } \\ { 4 x - y + 3 z = 8 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{317}{69} = 4\frac{41}{69} \approx 4.594202899$$

$$y=\frac{20}{69}\approx 0.289855072$$

$$z = -\frac{232}{69} = -3\frac{25}{69} \approx -3.362318841$$

Short Steps Using Substitution

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

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

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

$$5\left(3y-2z-3\right)-46y-z=13 4\left(3y-2z-3\right)-y+3z=8$$

Solve these equations for y and z respectively.

$$y=-\frac{11}{31}z-\frac{28}{31} z=\frac{11}{5}y-4$$

Substitute -\frac{11}{31}z-\frac{28}{31} for y in the equation z=\frac{11}{5}y-4.

$$z=\frac{11}{5}\left(-\frac{11}{31}z-\frac{28}{31}\right)-4$$

Solve z=\frac{11}{5}\left(-\frac{11}{31}z-\frac{28}{31}\right)-4 for z.

$$z=-\frac{232}{69}$$

Substitute -\frac{232}{69} for z in the equation y=-\frac{11}{31}z-\frac{28}{31}.

$$y=-\frac{11}{31}\left(-\frac{232}{69}\right)-\frac{28}{31}$$

Calculate y from y=-\frac{11}{31}\left(-\frac{232}{69}\right)-\frac{28}{31}.

$$y=\frac{20}{69}$$

Substitute \frac{20}{69} for y and -\frac{232}{69} for z in the equation x=3y-2z-3.

$$x=3\times \frac{20}{69}-2\left(-\frac{232}{69}\right)-3$$

Calculate x from x=3\times \frac{20}{69}-2\left(-\frac{232}{69}\right)-3.

$$x=\frac{317}{69}$$

The system is now solved.

$$x=\frac{317}{69} y=\frac{20}{69} z=-\frac{232}{69}$$