Solve 2(x-p)=3(6p-x):x

$$2 ( x - p ) = 3 ( 6 p - x ) : x$$

Solve for p

$$p=\frac{x\left(2x+3\right)}{2\left(x+9\right)}$$

$$x\neq -9\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$2\left(x-p\right)x=3\left(6p-x\right)$$

Use the distributive property to multiply 2 by x-p.

$$\left(2x-2p\right)x=3\left(6p-x\right)$$

Use the distributive property to multiply 2x-2p by x.

$$2x^{2}-2px=3\left(6p-x\right)$$

Use the distributive property to multiply 3 by 6p-x.

$$2x^{2}-2px=18p-3x$$

Subtract 18p from both sides.

$$2x^{2}-2px-18p=-3x$$

Subtract 2x^{2} from both sides.

$$-2px-18p=-3x-2x^{2}$$

Combine all terms containing p.

$$\left(-2x-18\right)p=-3x-2x^{2}$$

The equation is in standard form.

$$\left(-2x-18\right)p=-2x^{2}-3x$$

Divide both sides by -2x-18.

$$\frac{\left(-2x-18\right)p}{-2x-18}=-\frac{x\left(2x+3\right)}{-2x-18}$$

Dividing by -2x-18 undoes the multiplication by -2x-18.

$$p=-\frac{x\left(2x+3\right)}{-2x-18}$$

Divide -x\left(3+2x\right) by -2x-18.

$$p=\frac{x\left(2x+3\right)}{2\left(x+9\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-\frac{\sqrt{4p^{2}+132p+9}}{4}+\frac{p}{2}-\frac{3}{4}\text{, }&\text{unconditionally}\\x=\frac{\sqrt{4p^{2}+132p+9}}{4}+\frac{p}{2}-\frac{3}{4}\text{, }&p\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{4p^{2}+132p+9}}{4}+\frac{p}{2}-\frac{3}{4}\text{, }&p\geq 3\sqrt{30}-\frac{33}{2}\text{ or }p\leq -3\sqrt{30}-\frac{33}{2}\\x=\frac{\sqrt{4p^{2}+132p+9}}{4}+\frac{p}{2}-\frac{3}{4}\text{, }&\left(p\neq 0\text{ and }p\geq 3\sqrt{30}-\frac{33}{2}\right)\text{ or }p\leq -3\sqrt{30}-\frac{33}{2}\end{matrix}\right.$$