Solve 1/frac{1{A}+1/7}=P*A/Q+1

$$\frac { 1 } { \frac { 1 } { A } + \frac { 1 } { 7 } } = \frac { P \cdot A } { Q + 1 }$$

Solve for P

$$P=\frac{7\left(Q+1\right)}{A+7}$$

$$A\neq -7\text{ and }A\neq 0\text{ and }Q\neq -1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by Q+1.

$$7\left(Q+1\right)\left(7+A\right)^{-1}A=PA$$

Use the distributive property to multiply 7 by Q+1.

$$\left(7Q+7\right)\left(7+A\right)^{-1}A=PA$$

Use the distributive property to multiply 7Q+7 by \left(7+A\right)^{-1}.

$$\left(7Q\left(7+A\right)^{-1}+7\left(7+A\right)^{-1}\right)A=PA$$

Use the distributive property to multiply 7Q\left(7+A\right)^{-1}+7\left(7+A\right)^{-1} by A.

$$7Q\left(7+A\right)^{-1}A+7\left(7+A\right)^{-1}A=PA$$

Swap sides so that all variable terms are on the left hand side.

$$PA=7Q\left(7+A\right)^{-1}A+7\left(7+A\right)^{-1}A$$

Reorder the terms.

$$AP=7\times \frac{1}{A+7}AQ+7\times \frac{1}{A+7}A$$

Multiply both sides of the equation by A+7.

$$AP\left(A+7\right)=7\times 1AQ+7\times 1A$$

Use the distributive property to multiply AP by A+7.

$$PA^{2}+7AP=7\times 1AQ+7\times 1A$$

Do the multiplications.

$$PA^{2}+7AP=7AQ+7A$$

Combine all terms containing P.

$$\left(A^{2}+7A\right)P=7AQ+7A$$

Divide both sides by A^{2}+7A.

$$\frac{\left(A^{2}+7A\right)P}{A^{2}+7A}=\frac{7A\left(Q+1\right)}{A^{2}+7A}$$

Dividing by A^{2}+7A undoes the multiplication by A^{2}+7A.

$$P=\frac{7A\left(Q+1\right)}{A^{2}+7A}$$

Divide 7A\left(1+Q\right) by A^{2}+7A.

$$P=\frac{7\left(Q+1\right)}{A+7}$$

Solve for A

$$A=\frac{7\left(1+Q-P\right)}{P}$$

$$Q\neq -1\text{ and }P\neq 0\text{ and }P\neq Q+1$$