Solve {l}{3x+2y+5z=2}{5x-3y-2z=4}{2x-5y-3z=14}

$$\left. \begin{array} { l } { 3 x + 2 y + 5 z = 2 } \\ { 5 x - 3 y - 2 z = 4 } \\ { 2 x - 5 y - 3 z = 14 } \end{array} \right.$$

Solve for x, y, z

$$x=-1$$

$$y=-5$$

$$z=3$$

Short Steps Using Substitution

Solve 3x+2y+5z=2 for x.

$$x=-\frac{2}{3}y-\frac{5}{3}z+\frac{2}{3}$$

Substitute -\frac{2}{3}y-\frac{5}{3}z+\frac{2}{3} for x in the second and third equation.

$$5\left(-\frac{2}{3}y-\frac{5}{3}z+\frac{2}{3}\right)-3y-2z=4 2\left(-\frac{2}{3}y-\frac{5}{3}z+\frac{2}{3}\right)-5y-3z=14$$

Solve these equations for y and z respectively.

$$y=-\frac{2}{19}-\frac{31}{19}z z=-2-y$$

Substitute -\frac{2}{19}-\frac{31}{19}z for y in the equation z=-2-y.

$$z=-2-\left(-\frac{2}{19}-\frac{31}{19}z\right)$$

Solve z=-2-\left(-\frac{2}{19}-\frac{31}{19}z\right) for z.

$$z=3$$

Substitute 3 for z in the equation y=-\frac{2}{19}-\frac{31}{19}z.

$$y=-\frac{2}{19}-\frac{31}{19}\times 3$$

Calculate y from y=-\frac{2}{19}-\frac{31}{19}\times 3.

$$y=-5$$

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

$$x=-\frac{2}{3}\left(-5\right)-\frac{5}{3}\times 3+\frac{2}{3}$$

Calculate x from x=-\frac{2}{3}\left(-5\right)-\frac{5}{3}\times 3+\frac{2}{3}.

$$x=-1$$

The system is now solved.

$$x=-1 y=-5 z=3$$