Solve AB^2+AC^2=BC^2+31

$$A B ^ { 2 } + A C ^ { 2 } = B C ^ { 2 } + 31$$

Solve for A

$$A=\frac{BC^{2}+31}{B^{2}+C^{2}}$$

$$C\neq 0\text{ or }B\neq 0$$

Steps for Solving Linear Equation

Combine all terms containing A.

$$\left(B^{2}+C^{2}\right)A=BC^{2}+31$$

Divide both sides by B^{2}+C^{2}.

$$\frac{\left(B^{2}+C^{2}\right)A}{B^{2}+C^{2}}=\frac{BC^{2}+31}{B^{2}+C^{2}}$$

Dividing by B^{2}+C^{2} undoes the multiplication by B^{2}+C^{2}.

$$A=\frac{BC^{2}+31}{B^{2}+C^{2}}$$

Solve for B

$$\left\{\begin{matrix}B=\frac{\sqrt{124A+C^{4}-4\left(AC\right)^{2}}+C^{2}}{2A}\text{; }B=\frac{-\sqrt{124A+C^{4}-4\left(AC\right)^{2}}+C^{2}}{2A}\text{, }&\left(A\neq 0\text{ and }C=0\right)\text{ or }\left(A\neq 0\text{ and }A\geq -\frac{4\sqrt{C^{6}+961}-124}{8C^{2}}\text{ and }A\leq \frac{4\sqrt{C^{6}+961}+124}{8C^{2}}\right)\\B=-\frac{31}{C^{2}}\text{, }&C\neq 0\text{ and }A=0\end{matrix}\right.$$