Solve BP/PC*CQ/QA*AR/RB=1

$$\frac { B P } { P C } \cdot \frac { C Q } { Q A } \cdot \frac { A R } { R B } = 1$$

Solve for A

$$A\neq 0$$

$$B\neq 0\text{ and }P\neq 0\text{ and }C\neq 0\text{ and }Q\neq 0\text{ and }R\neq 0$$

Short Solution Steps

Variable A cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by ABCPQR, the least common multiple of PC,QA,RB.

$$ABQRBP\times \frac{CQ}{QA}\times \frac{AR}{RB}=ABCPQR$$

Multiply B and B to get B^{2}.

$$AB^{2}QRP\times \frac{CQ}{QA}\times \frac{AR}{RB}=ABCPQR$$

Cancel out Q in both numerator and denominator.

$$AB^{2}QRP\times \frac{C}{A}\times \frac{AR}{RB}=ABCPQR$$

Cancel out R in both numerator and denominator.

$$AB^{2}QRP\times \frac{C}{A}\times \frac{A}{B}=ABCPQR$$

Express A\times \frac{C}{A} as a single fraction.

$$\frac{AC}{A}B^{2}QRP\times \frac{A}{B}=ABCPQR$$

Multiply \frac{AC}{A} times \frac{A}{B} by multiplying numerator times numerator and denominator times denominator.

$$\frac{ACA}{AB}B^{2}QRP=ABCPQR$$

Multiply A and A to get A^{2}.

$$\frac{A^{2}C}{AB}B^{2}QRP=ABCPQR$$

Express \frac{A^{2}C}{AB}B^{2} as a single fraction.

$$\frac{A^{2}CB^{2}}{AB}QRP=ABCPQR$$

Cancel out B in both numerator and denominator.

$$\frac{BCA^{2}}{A}QRP=ABCPQR$$

Express \frac{BCA^{2}}{A}Q as a single fraction.

$$\frac{BCA^{2}Q}{A}RP=ABCPQR$$

Express \frac{BCA^{2}Q}{A}R as a single fraction.

$$\frac{BCA^{2}QR}{A}P=ABCPQR$$

Express \frac{BCA^{2}QR}{A}P as a single fraction.

$$\frac{BCA^{2}QRP}{A}=ABCPQR$$

Subtract ABCPQR from both sides.

$$\frac{BCA^{2}QRP}{A}-ABCPQR=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply ABCPQR times \frac{A}{A}.

$$\frac{BCA^{2}QRP}{A}-\frac{ABCPQRA}{A}=0$$

Since \frac{BCA^{2}QRP}{A} and \frac{ABCPQRA}{A} have the same denominator, subtract them by subtracting their numerators.

$$\frac{BCA^{2}QRP-ABCPQRA}{A}=0$$

Do the multiplications in BCA^{2}QRP-ABCPQRA.

$$\frac{BCA^{2}QRP-QA^{2}BCPR}{A}=0$$

Combine like terms in BCA^{2}QRP-QA^{2}BCPR.

$$\frac{0}{A}=0$$

Variable A cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by A.

$$0=0$$

Reorder the terms.

$$\text{true}$$

This is true for any A.

$$A\in \mathrm{R}$$

Variable A cannot be equal to 0.

$$A\in \mathrm{R}\setminus 0$$

Solve for B

$$B\neq 0$$

$$P\neq 0\text{ and }C\neq 0\text{ and }Q\neq 0\text{ and }A\neq 0\text{ and }R\neq 0$$

Short Solution Steps

Variable B cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by ABCPQR, the least common multiple of PC,QA,RB.

$$ABQRBP\times \frac{CQ}{QA}\times \frac{AR}{RB}=ABCPQR$$

Multiply B and B to get B^{2}.

$$AB^{2}QRP\times \frac{CQ}{QA}\times \frac{AR}{RB}=ABCPQR$$

Cancel out Q in both numerator and denominator.

$$AB^{2}QRP\times \frac{C}{A}\times \frac{AR}{RB}=ABCPQR$$

Cancel out R in both numerator and denominator.

$$AB^{2}QRP\times \frac{C}{A}\times \frac{A}{B}=ABCPQR$$

Express A\times \frac{C}{A} as a single fraction.

$$\frac{AC}{A}B^{2}QRP\times \frac{A}{B}=ABCPQR$$

Cancel out A in both numerator and denominator.

$$CB^{2}QRP\times \frac{A}{B}=ABCPQR$$

Express C\times \frac{A}{B} as a single fraction.

$$\frac{CA}{B}B^{2}QRP=ABCPQR$$

Express \frac{CA}{B}B^{2} as a single fraction.

$$\frac{CAB^{2}}{B}QRP=ABCPQR$$

Express \frac{CAB^{2}}{B}Q as a single fraction.

$$\frac{CAB^{2}Q}{B}RP=ABCPQR$$

Express \frac{CAB^{2}Q}{B}R as a single fraction.

$$\frac{CAB^{2}QR}{B}P=ABCPQR$$

Express \frac{CAB^{2}QR}{B}P as a single fraction.

$$\frac{CAB^{2}QRP}{B}=ABCPQR$$

Subtract ABCPQR from both sides.

$$\frac{CAB^{2}QRP}{B}-ABCPQR=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply ABCPQR times \frac{B}{B}.

$$\frac{CAB^{2}QRP}{B}-\frac{ABCPQRB}{B}=0$$

Since \frac{CAB^{2}QRP}{B} and \frac{ABCPQRB}{B} have the same denominator, subtract them by subtracting their numerators.

$$\frac{CAB^{2}QRP-ABCPQRB}{B}=0$$

Do the multiplications in CAB^{2}QRP-ABCPQRB.

$$\frac{CAB^{2}QRP-QAB^{2}CPR}{B}=0$$

Combine like terms in CAB^{2}QRP-QAB^{2}CPR.

$$\frac{0}{B}=0$$

Variable B cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by B.

$$0=0$$

Reorder the terms.

$$\text{true}$$

This is true for any B.

$$B\in \mathrm{R}$$

Variable B cannot be equal to 0.

$$B\in \mathrm{R}\setminus 0$$