Solve for A (complex solution)
\left\{\begin{matrix}A=\frac{BC}{2Q-C-B}\text{, }&Q\neq \frac{B+C}{2}\\A\in \mathrm{C}\text{, }&\left(C=0\text{ and }Q=\frac{B}{2}\right)\text{ or }\left(B=0\text{ and }Q=\frac{C}{2}\right)\end{matrix}\right.
Solve for B (complex solution)
\left\{\begin{matrix}B=-\frac{A\left(C-2Q\right)}{A+C}\text{, }&A\neq -C\\B\in \mathrm{C}\text{, }&\left(A=0\text{ and }C=0\right)\text{ or }\left(Q=\frac{C}{2}\text{ and }A=-C\right)\end{matrix}\right.
Solve for A
\left\{\begin{matrix}A=\frac{BC}{2Q-C-B}\text{, }&Q\neq \frac{B+C}{2}\\A\in \mathrm{R}\text{, }&\left(C=0\text{ and }Q=\frac{B}{2}\right)\text{ or }\left(B=0\text{ and }Q=\frac{C}{2}\right)\end{matrix}\right.
Solve for B
\left\{\begin{matrix}B=-\frac{A\left(C-2Q\right)}{A+C}\text{, }&A\neq -C\\B\in \mathrm{R}\text{, }&\left(A=0\text{ and }C=0\right)\text{ or }\left(Q=\frac{C}{2}\text{ and }A=-C\right)\end{matrix}\right.
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AQ=\frac{1}{2}BC+\frac{1}{2}CA+\frac{1}{2}AB
Use the distributive property to multiply \frac{1}{2} by BC+CA+AB.
AQ-\frac{1}{2}CA=\frac{1}{2}BC+\frac{1}{2}AB
Subtract \frac{1}{2}CA from both sides.
AQ-\frac{1}{2}CA-\frac{1}{2}AB=\frac{1}{2}BC
Subtract \frac{1}{2}AB from both sides.
\left(Q-\frac{1}{2}C-\frac{1}{2}B\right)A=\frac{1}{2}BC
Combine all terms containing A.
\left(-\frac{B}{2}-\frac{C}{2}+Q\right)A=\frac{BC}{2}
The equation is in standard form.
\frac{\left(-\frac{B}{2}-\frac{C}{2}+Q\right)A}{-\frac{B}{2}-\frac{C}{2}+Q}=\frac{BC}{2\left(-\frac{B}{2}-\frac{C}{2}+Q\right)}
Divide both sides by Q-\frac{1}{2}C-\frac{1}{2}B.
A=\frac{BC}{2\left(-\frac{B}{2}-\frac{C}{2}+Q\right)}
Dividing by Q-\frac{1}{2}C-\frac{1}{2}B undoes the multiplication by Q-\frac{1}{2}C-\frac{1}{2}B.
A=\frac{BC}{2Q-C-B}
Divide \frac{BC}{2} by Q-\frac{1}{2}C-\frac{1}{2}B.
AQ=\frac{1}{2}BC+\frac{1}{2}CA+\frac{1}{2}AB
Use the distributive property to multiply \frac{1}{2} by BC+CA+AB.
\frac{1}{2}BC+\frac{1}{2}CA+\frac{1}{2}AB=AQ
Swap sides so that all variable terms are on the left hand side.
\frac{1}{2}BC+\frac{1}{2}AB=AQ-\frac{1}{2}CA
Subtract \frac{1}{2}CA from both sides.
\left(\frac{1}{2}C+\frac{1}{2}A\right)B=AQ-\frac{1}{2}CA
Combine all terms containing B.
\frac{A+C}{2}B=-\frac{AC}{2}+AQ
The equation is in standard form.
\frac{2\times \frac{A+C}{2}B}{A+C}=\frac{2\left(-\frac{AC}{2}+AQ\right)}{A+C}
Divide both sides by \frac{1}{2}C+\frac{1}{2}A.
B=\frac{2\left(-\frac{AC}{2}+AQ\right)}{A+C}
Dividing by \frac{1}{2}C+\frac{1}{2}A undoes the multiplication by \frac{1}{2}C+\frac{1}{2}A.
B=\frac{A\left(2Q-C\right)}{A+C}
Divide AQ-\frac{AC}{2} by \frac{1}{2}C+\frac{1}{2}A.
AQ=\frac{1}{2}BC+\frac{1}{2}CA+\frac{1}{2}AB
Use the distributive property to multiply \frac{1}{2} by BC+CA+AB.
AQ-\frac{1}{2}CA=\frac{1}{2}BC+\frac{1}{2}AB
Subtract \frac{1}{2}CA from both sides.
AQ-\frac{1}{2}CA-\frac{1}{2}AB=\frac{1}{2}BC
Subtract \frac{1}{2}AB from both sides.
\left(Q-\frac{1}{2}C-\frac{1}{2}B\right)A=\frac{1}{2}BC
Combine all terms containing A.
\left(-\frac{B}{2}-\frac{C}{2}+Q\right)A=\frac{BC}{2}
The equation is in standard form.
\frac{\left(-\frac{B}{2}-\frac{C}{2}+Q\right)A}{-\frac{B}{2}-\frac{C}{2}+Q}=\frac{BC}{2\left(-\frac{B}{2}-\frac{C}{2}+Q\right)}
Divide both sides by Q-\frac{1}{2}C-\frac{1}{2}B.
A=\frac{BC}{2\left(-\frac{B}{2}-\frac{C}{2}+Q\right)}
Dividing by Q-\frac{1}{2}C-\frac{1}{2}B undoes the multiplication by Q-\frac{1}{2}C-\frac{1}{2}B.
A=\frac{BC}{2Q-C-B}
Divide \frac{BC}{2} by Q-\frac{1}{2}C-\frac{1}{2}B.
AQ=\frac{1}{2}BC+\frac{1}{2}CA+\frac{1}{2}AB
Use the distributive property to multiply \frac{1}{2} by BC+CA+AB.
\frac{1}{2}BC+\frac{1}{2}CA+\frac{1}{2}AB=AQ
Swap sides so that all variable terms are on the left hand side.
\frac{1}{2}BC+\frac{1}{2}AB=AQ-\frac{1}{2}CA
Subtract \frac{1}{2}CA from both sides.
\left(\frac{1}{2}C+\frac{1}{2}A\right)B=AQ-\frac{1}{2}CA
Combine all terms containing B.
\frac{A+C}{2}B=-\frac{AC}{2}+AQ
The equation is in standard form.
\frac{2\times \frac{A+C}{2}B}{A+C}=\frac{2\left(-\frac{AC}{2}+AQ\right)}{A+C}
Divide both sides by \frac{1}{2}C+\frac{1}{2}A.
B=\frac{2\left(-\frac{AC}{2}+AQ\right)}{A+C}
Dividing by \frac{1}{2}C+\frac{1}{2}A undoes the multiplication by \frac{1}{2}C+\frac{1}{2}A.
B=\frac{A\left(2Q-C\right)}{A+C}
Divide AQ-\frac{AC}{2} by \frac{1}{2}C+\frac{1}{2}A.
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