Solve -3km/1+3k^2=-k-m/k^2+1

$$\frac { - 3 k m } { 1 + 3 k ^ { 2 } } = \frac { - k - m } { k ^ { 2 } + 1 }$$

Solve for m

$$m=-\frac{k\left(3k^{2}+1\right)}{1-3k+3k^{2}-3k^{3}}$$

$$k\neq \frac{\sqrt[3]{4}+1-\sqrt[3]{2}}{3}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(3k^{2}+1\right)\left(k^{2}+1\right), the least common multiple of 1+3k^{2},k^{2}+1.

$$\left(k^{2}+1\right)\left(-3\right)km=\left(3k^{2}+1\right)\left(-k-m\right)$$

Use the distributive property to multiply k^{2}+1 by -3.

$$\left(-3k^{2}-3\right)km=\left(3k^{2}+1\right)\left(-k-m\right)$$

Use the distributive property to multiply -3k^{2}-3 by k.

$$\left(-3k^{3}-3k\right)m=\left(3k^{2}+1\right)\left(-k-m\right)$$

Use the distributive property to multiply -3k^{3}-3k by m.

$$-3k^{3}m-3km=\left(3k^{2}+1\right)\left(-k-m\right)$$

Use the distributive property to multiply 3k^{2}+1 by -k-m.

$$-3k^{3}m-3km=3k^{2}\left(-k\right)-3k^{2}m-k-m$$

Add 3k^{2}m to both sides.

$$-3k^{3}m-3km+3k^{2}m=3k^{2}\left(-k\right)-k-m$$

Add m to both sides.

$$-3k^{3}m-3km+3k^{2}m+m=3k^{2}\left(-k\right)-k$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$-3k^{3}m-3km+3k^{2}m+m=3k^{3}\left(-1\right)-k$$

Multiply 3 and -1 to get -3.

$$-3k^{3}m-3km+3k^{2}m+m=-3k^{3}-k$$

Combine all terms containing m.

$$\left(-3k^{3}-3k+3k^{2}+1\right)m=-3k^{3}-k$$

The equation is in standard form.

$$\left(1-3k+3k^{2}-3k^{3}\right)m=-3k^{3}-k$$

Divide both sides by -3k^{3}+3k^{2}-3k+1.

$$\frac{\left(1-3k+3k^{2}-3k^{3}\right)m}{1-3k+3k^{2}-3k^{3}}=-\frac{k\left(3k^{2}+1\right)}{1-3k+3k^{2}-3k^{3}}$$

Dividing by -3k^{3}+3k^{2}-3k+1 undoes the multiplication by -3k^{3}+3k^{2}-3k+1.

$$m=-\frac{k\left(3k^{2}+1\right)}{1-3k+3k^{2}-3k^{3}}$$