Solve {l}{3x+4y=11}{5x+6z=35}{7y+8z=61}

$$\left. \begin{array} { l } { 3 x + 4 y = 11 } \\ { 5 x + 6 z = 35 } \\ { 7 y + 8 z = 61 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{59}{143}\approx 0.412587413$$

$$y = \frac{349}{143} = 2\frac{63}{143} \approx 2.440559441$$

$$z = \frac{785}{143} = 5\frac{70}{143} \approx 5.48951049$$

Short Steps Using Substitution

Solve 3x+4y=11 for x.

$$x=-\frac{4}{3}y+\frac{11}{3}$$

Substitute -\frac{4}{3}y+\frac{11}{3} for x in the equation 5x+6z=35.

$$5\left(-\frac{4}{3}y+\frac{11}{3}\right)+6z=35$$

Solve the second equation for y and the third equation for z.

$$y=-\frac{5}{2}+\frac{9}{10}z z=-\frac{7}{8}y+\frac{61}{8}$$

Substitute -\frac{5}{2}+\frac{9}{10}z for y in the equation z=-\frac{7}{8}y+\frac{61}{8}.

$$z=-\frac{7}{8}\left(-\frac{5}{2}+\frac{9}{10}z\right)+\frac{61}{8}$$

Solve z=-\frac{7}{8}\left(-\frac{5}{2}+\frac{9}{10}z\right)+\frac{61}{8} for z.

$$z=\frac{785}{143}$$

Substitute \frac{785}{143} for z in the equation y=-\frac{5}{2}+\frac{9}{10}z.

$$y=-\frac{5}{2}+\frac{9}{10}\times \frac{785}{143}$$

Calculate y from y=-\frac{5}{2}+\frac{9}{10}\times \frac{785}{143}.

$$y=\frac{349}{143}$$

Substitute \frac{349}{143} for y in the equation x=-\frac{4}{3}y+\frac{11}{3}.

$$x=-\frac{4}{3}\times \frac{349}{143}+\frac{11}{3}$$

Calculate x from x=-\frac{4}{3}\times \frac{349}{143}+\frac{11}{3}.

$$x=\frac{59}{143}$$

The system is now solved.

$$x=\frac{59}{143} y=\frac{349}{143} z=\frac{785}{143}$$