Solve {l}{4n-2m-3r=1}{m+3n-5r=-4}{3m-5n+r=0}

$$\left\{ \begin{array} { l } { 4 n - 2 m - 3 r = 1 } \\ { m + 3 n - 5 r = - 4 } \\ { 3 m - 5 n + r = 0 } \end{array} \right.$$

Solve for n, m, r

$$r=-1$$

$$n=-2$$

$$m=-3$$

Short Steps Using Substitution

Reorder the equations.

$$m+3n-5r=-4 4n-2m-3r=1 3m-5n+r=0$$

Solve m+3n-5r=-4 for m.

$$m=-3n+5r-4$$

Substitute -3n+5r-4 for m in the second and third equation.

$$4n-2\left(-3n+5r-4\right)-3r=1 3\left(-3n+5r-4\right)-5n+r=0$$

Solve these equations for n and r respectively.

$$n=\frac{13}{10}r-\frac{7}{10} r=\frac{3}{4}+\frac{7}{8}n$$

Substitute \frac{13}{10}r-\frac{7}{10} for n in the equation r=\frac{3}{4}+\frac{7}{8}n.

$$r=\frac{3}{4}+\frac{7}{8}\left(\frac{13}{10}r-\frac{7}{10}\right)$$

Solve r=\frac{3}{4}+\frac{7}{8}\left(\frac{13}{10}r-\frac{7}{10}\right) for r.

$$r=-1$$

Substitute -1 for r in the equation n=\frac{13}{10}r-\frac{7}{10}.

$$n=\frac{13}{10}\left(-1\right)-\frac{7}{10}$$

Calculate n from n=\frac{13}{10}\left(-1\right)-\frac{7}{10}.

$$n=-2$$

Substitute -2 for n and -1 for r in the equation m=-3n+5r-4.

$$m=-3\left(-2\right)+5\left(-1\right)-4$$

Calculate m from m=-3\left(-2\right)+5\left(-1\right)-4.

$$m=-3$$

The system is now solved.

$$n=-2 m=-3 r=-1$$