Solve frac{6+18P_{2}}{3R}+frac{4+8P_{2}}{4P_{2}}=17

$$\frac { 6 + 18 P _ { 2 } } { 3 R } + \frac { 4 + 8 P _ { 2 } } { 4 P _ { 2 } } = 17$$

Solve for R

$$R=-\frac{2P_{2}\left(3P_{2}+1\right)}{1-15P_{2}}$$

$$P_{2}\neq -\frac{1}{3}\text{ and }P_{2}\neq 0\text{ and }P_{2}\neq \frac{1}{15}$$

Steps for Solving Linear Equation

Variable R cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 12P_{2}R, the least common multiple of 3R,4P_{2}.

$$4P_{2}\left(6+18P_{2}\right)+3R\left(4+8P_{2}\right)=204P_{2}R$$

Use the distributive property to multiply 4P_{2} by 6+18P_{2}.

$$24P_{2}+72P_{2}^{2}+3R\left(4+8P_{2}\right)=204P_{2}R$$

Use the distributive property to multiply 3R by 4+8P_{2}.

$$24P_{2}+72P_{2}^{2}+12R+24RP_{2}=204P_{2}R$$

Subtract 204P_{2}R from both sides.

$$24P_{2}+72P_{2}^{2}+12R+24RP_{2}-204P_{2}R=0$$

Combine 24RP_{2} and -204P_{2}R to get -180RP_{2}.

$$24P_{2}+72P_{2}^{2}+12R-180RP_{2}=0$$

Subtract 24P_{2} from both sides. Anything subtracted from zero gives its negation.

$$72P_{2}^{2}+12R-180RP_{2}=-24P_{2}$$

Subtract 72P_{2}^{2} from both sides.

$$12R-180RP_{2}=-24P_{2}-72P_{2}^{2}$$

Combine all terms containing R.

$$\left(12-180P_{2}\right)R=-24P_{2}-72P_{2}^{2}$$

The equation is in standard form.

$$\left(12-180P_{2}\right)R=-72P_{2}^{2}-24P_{2}$$

Divide both sides by 12-180P_{2}.

$$\frac{\left(12-180P_{2}\right)R}{12-180P_{2}}=-\frac{24P_{2}\left(3P_{2}+1\right)}{12-180P_{2}}$$

Dividing by 12-180P_{2} undoes the multiplication by 12-180P_{2}.

$$R=-\frac{24P_{2}\left(3P_{2}+1\right)}{12-180P_{2}}$$

Divide -24P_{2}\left(1+3P_{2}\right) by 12-180P_{2}.

$$R=-\frac{2P_{2}\left(3P_{2}+1\right)}{1-15P_{2}}$$

Variable R cannot be equal to 0.

$$R=-\frac{2P_{2}\left(3P_{2}+1\right)}{1-15P_{2}}\text{, }R\neq 0$$

Solve for P_2

$$P_{2}=\frac{\sqrt{225R^{2}-84R+4}}{12}+\frac{5R}{4}-\frac{1}{6}$$

$$P_{2}=-\frac{\sqrt{225R^{2}-84R+4}}{12}+\frac{5R}{4}-\frac{1}{6}\text{, }\left(R\neq 0\text{ and }R\leq \frac{14-4\sqrt{6}}{75}\right)\text{ or }R\geq \frac{4\sqrt{6}+14}{75}$$