Solve {l}{5x-y-2=0}{x+2y+3z=14}{4x+3y+2z=16}

$$\left\{ \begin{array} { l } { 5 x - y - 2 = 0 } \\ { x + 2 y + 3 z = 14 } \\ { 4 x + 3 y + 2 z = 16 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{6}{7}\approx 0.857142857$$

$$y = \frac{16}{7} = 2\frac{2}{7} \approx 2.285714286$$

$$z = \frac{20}{7} = 2\frac{6}{7} \approx 2.857142857$$

Short Steps Using Substitution

Solve 5x-y-2=0 for y.

$$y=5x-2$$

Substitute 5x-2 for y in the second and third equation.

$$x+2\left(5x-2\right)+3z=14 4x+3\left(5x-2\right)+2z=16$$

Solve these equations for x and z respectively.

$$x=\frac{18}{11}-\frac{3}{11}z z=-\frac{19}{2}x+11$$

Substitute \frac{18}{11}-\frac{3}{11}z for x in the equation z=-\frac{19}{2}x+11.

$$z=-\frac{19}{2}\left(\frac{18}{11}-\frac{3}{11}z\right)+11$$

Solve z=-\frac{19}{2}\left(\frac{18}{11}-\frac{3}{11}z\right)+11 for z.

$$z=\frac{20}{7}$$

Substitute \frac{20}{7} for z in the equation x=\frac{18}{11}-\frac{3}{11}z.

$$x=\frac{18}{11}-\frac{3}{11}\times \frac{20}{7}$$

Calculate x from x=\frac{18}{11}-\frac{3}{11}\times \frac{20}{7}.

$$x=\frac{6}{7}$$

Substitute \frac{6}{7} for x in the equation y=5x-2.

$$y=5\times \frac{6}{7}-2$$

Calculate y from y=5\times \frac{6}{7}-2.

$$y=\frac{16}{7}$$

The system is now solved.

$$x=\frac{6}{7} y=\frac{16}{7} z=\frac{20}{7}$$