Solve frac{v_{1}-12}{3k}+frac{v_{1}}{2k}+v_{1}+v_{2}=0

$$\frac { v _ { 1 } - 12 } { 3 k } + \frac { v _ { 1 } } { 2 k } + v _ { 1 } + v _ { 2 } = 0$$

Solve for v_1 (complex solution)

$$\left\{\begin{matrix}v_{1}=-\frac{6\left(kv_{2}-4\right)}{6k+5}\text{, }&k\neq -\frac{5}{6}\text{ and }k\neq 0\\v_{1}\in \mathrm{C}\text{, }&v_{2}=-\frac{24}{5}\text{ and }k=-\frac{5}{6}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6k, the least common multiple of 3k,2k.

$$2\left(v_{1}-12\right)+3v_{1}+6kv_{1}+6kv_{2}=0$$

Use the distributive property to multiply 2 by v_{1}-12.

$$2v_{1}-24+3v_{1}+6kv_{1}+6kv_{2}=0$$

Combine 2v_{1} and 3v_{1} to get 5v_{1}.

$$5v_{1}-24+6kv_{1}+6kv_{2}=0$$

Add 24 to both sides. Anything plus zero gives itself.

$$5v_{1}+6kv_{1}+6kv_{2}=24$$

Subtract 6kv_{2} from both sides.

$$5v_{1}+6kv_{1}=24-6kv_{2}$$

Combine all terms containing v_{1}.

$$\left(5+6k\right)v_{1}=24-6kv_{2}$$

The equation is in standard form.

$$\left(6k+5\right)v_{1}=24-6kv_{2}$$

Divide both sides by 6k+5.

$$\frac{\left(6k+5\right)v_{1}}{6k+5}=\frac{24-6kv_{2}}{6k+5}$$

Dividing by 6k+5 undoes the multiplication by 6k+5.

$$v_{1}=\frac{24-6kv_{2}}{6k+5}$$

Divide 24-6kv_{2} by 6k+5.

$$v_{1}=\frac{6\left(4-kv_{2}\right)}{6k+5}$$

Solve for k

$$\left\{\begin{matrix}k=-\frac{5v_{1}-24}{6\left(v_{1}+v_{2}\right)}\text{, }&v_{1}\neq \frac{24}{5}\text{ and }v_{1}\neq -v_{2}\\k\neq 0\text{, }&v_{1}=\frac{24}{5}\text{ and }v_{2}=-\frac{24}{5}\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable k cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 6k, the least common multiple of 3k,2k.

$$2\left(v_{1}-12\right)+3v_{1}+6kv_{1}+6kv_{2}=0$$

Use the distributive property to multiply 2 by v_{1}-12.

$$2v_{1}-24+3v_{1}+6kv_{1}+6kv_{2}=0$$

Combine 2v_{1} and 3v_{1} to get 5v_{1}.

$$5v_{1}-24+6kv_{1}+6kv_{2}=0$$

Subtract 5v_{1} from both sides. Anything subtracted from zero gives its negation.

$$-24+6kv_{1}+6kv_{2}=-5v_{1}$$

Add 24 to both sides.

$$6kv_{1}+6kv_{2}=-5v_{1}+24$$

Combine all terms containing k.

$$\left(6v_{1}+6v_{2}\right)k=-5v_{1}+24$$

The equation is in standard form.

$$\left(6v_{1}+6v_{2}\right)k=24-5v_{1}$$

Divide both sides by 6v_{2}+6v_{1}.

$$\frac{\left(6v_{1}+6v_{2}\right)k}{6v_{1}+6v_{2}}=\frac{24-5v_{1}}{6v_{1}+6v_{2}}$$

Dividing by 6v_{2}+6v_{1} undoes the multiplication by 6v_{2}+6v_{1}.

$$k=\frac{24-5v_{1}}{6v_{1}+6v_{2}}$$

Divide -5v_{1}+24 by 6v_{2}+6v_{1}.

$$k=\frac{24-5v_{1}}{6\left(v_{1}+v_{2}\right)}$$

Variable k cannot be equal to 0.

$$k=\frac{24-5v_{1}}{6\left(v_{1}+v_{2}\right)}\text{, }k\neq 0$$

Solve for v_1

$$\left\{\begin{matrix}v_{1}=-\frac{6\left(kv_{2}-4\right)}{6k+5}\text{, }&k\neq -\frac{5}{6}\text{ and }k\neq 0\\v_{1}\in \mathrm{R}\text{, }&v_{2}=-\frac{24}{5}\text{ and }k=-\frac{5}{6}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 6k, the least common multiple of 3k,2k.

$$2\left(v_{1}-12\right)+3v_{1}+6kv_{1}+6kv_{2}=0$$

Use the distributive property to multiply 2 by v_{1}-12.

$$2v_{1}-24+3v_{1}+6kv_{1}+6kv_{2}=0$$

Combine 2v_{1} and 3v_{1} to get 5v_{1}.

$$5v_{1}-24+6kv_{1}+6kv_{2}=0$$

Add 24 to both sides. Anything plus zero gives itself.

$$5v_{1}+6kv_{1}+6kv_{2}=24$$

Subtract 6kv_{2} from both sides.

$$5v_{1}+6kv_{1}=24-6kv_{2}$$

Combine all terms containing v_{1}.

$$\left(5+6k\right)v_{1}=24-6kv_{2}$$

The equation is in standard form.

$$\left(6k+5\right)v_{1}=24-6kv_{2}$$

Divide both sides by 6k+5.

$$\frac{\left(6k+5\right)v_{1}}{6k+5}=\frac{24-6kv_{2}}{6k+5}$$

Dividing by 6k+5 undoes the multiplication by 6k+5.

$$v_{1}=\frac{24-6kv_{2}}{6k+5}$$

Divide 24-6kv_{2} by 6k+5.

$$v_{1}=\frac{6\left(4-kv_{2}\right)}{6k+5}$$