Solve {l}{1+3r+2s=13-2t}{5r-s=-21+3t}{2+8r+14s=14-t}

$$\left\{ \begin{array} { l } { 1 + 3 r + 2 s = 13 - 2 t } \\ { 5 r - s = - 21 + 3 t } \\ { 2 + 8 r + 14 s = 14 - t } \end{array} \right.$$

Solve for r, s, t

$$r=-\frac{6}{13}\approx -0.461538462$$

$$t=6$$

$$s=\frac{9}{13}\approx 0.692307692$$

Short Steps Using Substitution

Reorder the equations.

$$5r-s=-21+3t 1+3r+2s=13-2t 2+8r+14s=14-t$$

Solve 5r-s=-21+3t for s.

$$s=5r+21-3t$$

Substitute 5r+21-3t for s in the second and third equation.

$$1+3r+2\left(5r+21-3t\right)=13-2t 2+8r+14\left(5r+21-3t\right)=14-t$$

Solve these equations for r and t respectively.

$$r=-\frac{30}{13}+\frac{4}{13}t t=\frac{282}{41}+\frac{78}{41}r$$

Substitute -\frac{30}{13}+\frac{4}{13}t for r in the equation t=\frac{282}{41}+\frac{78}{41}r.

$$t=\frac{282}{41}+\frac{78}{41}\left(-\frac{30}{13}+\frac{4}{13}t\right)$$

Solve t=\frac{282}{41}+\frac{78}{41}\left(-\frac{30}{13}+\frac{4}{13}t\right) for t.

$$t=6$$

Substitute 6 for t in the equation r=-\frac{30}{13}+\frac{4}{13}t.

$$r=-\frac{30}{13}+\frac{4}{13}\times 6$$

Calculate r from r=-\frac{30}{13}+\frac{4}{13}\times 6.

$$r=-\frac{6}{13}$$

Substitute -\frac{6}{13} for r and 6 for t in the equation s=5r+21-3t.

$$s=5\left(-\frac{6}{13}\right)+21-3\times 6$$

Calculate s from s=5\left(-\frac{6}{13}\right)+21-3\times 6.

$$s=\frac{9}{13}$$

The system is now solved.

$$r=-\frac{6}{13} s=\frac{9}{13} t=6$$