Solve {l}{3z+7x+5y=8}{2z+5x+11y=9}{6z+11x+23y=19}

$$\left\{ \begin{array} { l } { 3 z + 7 x + 5 y = 8 } \\ { 2 z + 5 x + 11 y = 9 } \\ { 6 z + 11 x + 23 y = 19 } \end{array} \right.$$

Solve for z, x, y

$$x=\frac{37}{41}\approx 0.902439024$$

$$y=\frac{18}{41}\approx 0.43902439$$

$$z=-\frac{7}{41}\approx -0.170731707$$

Short Steps Using Substitution

Solve 3z+7x+5y=8 for z.

$$z=-\frac{7}{3}x-\frac{5}{3}y+\frac{8}{3}$$

Substitute -\frac{7}{3}x-\frac{5}{3}y+\frac{8}{3} for z in the second and third equation.

$$2\left(-\frac{7}{3}x-\frac{5}{3}y+\frac{8}{3}\right)+5x+11y=9 6\left(-\frac{7}{3}x-\frac{5}{3}y+\frac{8}{3}\right)+11x+23y=19$$

Solve these equations for x and y respectively.

$$x=11-23y y=\frac{3}{13}+\frac{3}{13}x$$

Substitute 11-23y for x in the equation y=\frac{3}{13}+\frac{3}{13}x.

$$y=\frac{3}{13}+\frac{3}{13}\left(11-23y\right)$$

Solve y=\frac{3}{13}+\frac{3}{13}\left(11-23y\right) for y.

$$y=\frac{18}{41}$$

Substitute \frac{18}{41} for y in the equation x=11-23y.

$$x=11-23\times \frac{18}{41}$$

Calculate x from x=11-23\times \frac{18}{41}.

$$x=\frac{37}{41}$$

Substitute \frac{37}{41} for x and \frac{18}{41} for y in the equation z=-\frac{7}{3}x-\frac{5}{3}y+\frac{8}{3}.

$$z=-\frac{7}{3}\times \frac{37}{41}-\frac{5}{3}\times \frac{18}{41}+\frac{8}{3}$$

Calculate z from z=-\frac{7}{3}\times \frac{37}{41}-\frac{5}{3}\times \frac{18}{41}+\frac{8}{3}.

$$z=-\frac{7}{41}$$

The system is now solved.

$$z=-\frac{7}{41} x=\frac{37}{41} y=\frac{18}{41}$$