Solve {l}{6r-s+t=-13}{4r+2s-3t=-10}{r-3s+2t=-9}

$$\left. \begin{array}{l}{ 6 r - s + t = - 13 }\\{ 4 r + 2 s - 3 t = - 10 }\\{ r - 3 s + 2 t = - 9 }\end{array} \right.$$

Solve for r, s, t

$$r=-2$$

$$t=4$$

$$s=5$$

Short Steps Using Substitution

Solve 6r-s+t=-13 for s.

$$s=6r+t+13$$

Substitute 6r+t+13 for s in the second and third equation.

$$4r+2\left(6r+t+13\right)-3t=-10 r-3\left(6r+t+13\right)+2t=-9$$

Solve these equations for r and t respectively.

$$r=-\frac{9}{4}+\frac{1}{16}t t=-17r-30$$

Substitute -\frac{9}{4}+\frac{1}{16}t for r in the equation t=-17r-30.

$$t=-17\left(-\frac{9}{4}+\frac{1}{16}t\right)-30$$

Solve t=-17\left(-\frac{9}{4}+\frac{1}{16}t\right)-30 for t.

$$t=4$$

Substitute 4 for t in the equation r=-\frac{9}{4}+\frac{1}{16}t.

$$r=-\frac{9}{4}+\frac{1}{16}\times 4$$

Calculate r from r=-\frac{9}{4}+\frac{1}{16}\times 4.

$$r=-2$$

Substitute -2 for r and 4 for t in the equation s=6r+t+13.

$$s=6\left(-2\right)+4+13$$

Calculate s from s=6\left(-2\right)+4+13.

$$s=5$$

The system is now solved.

$$r=-2 s=5 t=4$$