Solve {l}{13x-y=23}{2x+3y-z=14}{y-2z=-5}

$$\left. \begin{array} { l } { 13 x - y = 23 } \\ { 2 x + 3 y - z = 14 } \\ { y - 2 z = - 5 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{148}{69} = 2\frac{10}{69} \approx 2.144927536$$

$$y = \frac{337}{69} = 4\frac{61}{69} \approx 4.884057971$$

$$z = \frac{341}{69} = 4\frac{65}{69} \approx 4.942028986$$

Short Steps Using Substitution

Solve 13x-y=23 for y.

$$y=13x-23$$

Substitute 13x-23 for y in the second and third equation.

$$2x+3\left(13x-23\right)-z=14 13x-23-2z=-5$$

Solve these equations for x and z respectively.

$$x=\frac{83}{41}+\frac{1}{41}z z=\frac{13}{2}x-9$$

Substitute \frac{83}{41}+\frac{1}{41}z for x in the equation z=\frac{13}{2}x-9.

$$z=\frac{13}{2}\left(\frac{83}{41}+\frac{1}{41}z\right)-9$$

Solve z=\frac{13}{2}\left(\frac{83}{41}+\frac{1}{41}z\right)-9 for z.

$$z=\frac{341}{69}$$

Substitute \frac{341}{69} for z in the equation x=\frac{83}{41}+\frac{1}{41}z.

$$x=\frac{83}{41}+\frac{1}{41}\times \frac{341}{69}$$

Calculate x from x=\frac{83}{41}+\frac{1}{41}\times \frac{341}{69}.

$$x=\frac{148}{69}$$

Substitute \frac{148}{69} for x in the equation y=13x-23.

$$y=13\times \frac{148}{69}-23$$

Calculate y from y=13\times \frac{148}{69}-23.

$$y=\frac{337}{69}$$

The system is now solved.

$$x=\frac{148}{69} y=\frac{337}{69} z=\frac{341}{69}$$