Solve {l}{2x+3y+z=15}{2x+10y+2z=23}{x-y-3z=-16}

$$\left\{ \begin{array} { l } { 2 x + 3 y + z = 15 } \\ { 2 x + 10 y + 2 z = 23 } \\ { x - y - 3 z = - 16 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{43}{11} = 3\frac{10}{11} \approx 3.909090909$$

$$y=\frac{9}{44}\approx 0.204545455$$

$$z = \frac{289}{44} = 6\frac{25}{44} \approx 6.568181818$$

Short Steps Using Substitution

Solve 2x+3y+z=15 for z.

$$z=-2x-3y+15$$

Substitute -2x-3y+15 for z in the second and third equation.

$$2x+10y+2\left(-2x-3y+15\right)=23 x-y-3\left(-2x-3y+15\right)=-16$$

Solve these equations for y and x respectively.

$$y=\frac{1}{2}x-\frac{7}{4} x=-\frac{8}{7}y+\frac{29}{7}$$

Substitute \frac{1}{2}x-\frac{7}{4} for y in the equation x=-\frac{8}{7}y+\frac{29}{7}.

$$x=-\frac{8}{7}\left(\frac{1}{2}x-\frac{7}{4}\right)+\frac{29}{7}$$

Solve x=-\frac{8}{7}\left(\frac{1}{2}x-\frac{7}{4}\right)+\frac{29}{7} for x.

$$x=\frac{43}{11}$$

Substitute \frac{43}{11} for x in the equation y=\frac{1}{2}x-\frac{7}{4}.

$$y=\frac{1}{2}\times \frac{43}{11}-\frac{7}{4}$$

Calculate y from y=\frac{1}{2}\times \frac{43}{11}-\frac{7}{4}.

$$y=\frac{9}{44}$$

Substitute \frac{9}{44} for y and \frac{43}{11} for x in the equation z=-2x-3y+15.

$$z=-2\times \frac{43}{11}-3\times \frac{9}{44}+15$$

Calculate z from z=-2\times \frac{43}{11}-3\times \frac{9}{44}+15.

$$z=\frac{289}{44}$$

The system is now solved.

$$x=\frac{43}{11} y=\frac{9}{44} z=\frac{289}{44}$$