Solve 2GMm/(r+h)^2=GMm/r^2

$$2 G \frac { M m } { ( r + h ) ^ { 2 } } = G \frac { M m } { r ^ { 2 } }$$

Solve for G

$$\left\{\begin{matrix}G=0\text{, }&r\neq -h\text{ and }r\neq 0\\G\in \mathrm{R}\text{, }&\left(r=-\sqrt{2}h+h\text{ and }h\neq 0\right)\text{ or }\left(r=\sqrt{2}h+h\text{ and }h\neq 0\right)\text{ or }\left(m=0\text{ and }r\neq -h\text{ and }r\neq 0\right)\text{ or }\left(M=0\text{ and }r\neq -h\text{ and }r\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by r^{2}\left(r+h\right)^{2}, the least common multiple of \left(r+h\right)^{2},r^{2}.

$$2Gr^{2}Mm=G\left(r+h\right)^{2}Mm$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(r+h\right)^{2}.

$$2Gr^{2}Mm=G\left(r^{2}+2rh+h^{2}\right)Mm$$

Use the distributive property to multiply G by r^{2}+2rh+h^{2}.

$$2Gr^{2}Mm=\left(Gr^{2}+2Grh+Gh^{2}\right)Mm$$

Use the distributive property to multiply Gr^{2}+2Grh+Gh^{2} by M.

$$2Gr^{2}Mm=\left(Gr^{2}M+2GrhM+Gh^{2}M\right)m$$

Use the distributive property to multiply Gr^{2}M+2GrhM+Gh^{2}M by m.

$$2Gr^{2}Mm=Gr^{2}Mm+2GrhMm+Gh^{2}Mm$$

Subtract Gr^{2}Mm from both sides.

$$2Gr^{2}Mm-Gr^{2}Mm=2GrhMm+Gh^{2}Mm$$

Combine 2Gr^{2}Mm and -Gr^{2}Mm to get Gr^{2}Mm.

$$Gr^{2}Mm=2GrhMm+Gh^{2}Mm$$

Subtract 2GrhMm from both sides.

$$Gr^{2}Mm-2GrhMm=Gh^{2}Mm$$

Subtract Gh^{2}Mm from both sides.

$$Gr^{2}Mm-2GrhMm-Gh^{2}Mm=0$$

Reorder the terms.

$$GMmr^{2}-2GMhmr-GMmh^{2}=0$$

Combine all terms containing G.

$$\left(Mmr^{2}-2Mhmr-Mmh^{2}\right)G=0$$

Divide 0 by Mmr^{2}-2Mhmr-Mmh^{2}.

$$G=0$$

Solve for M

$$\left\{\begin{matrix}M=0\text{, }&r\neq -h\text{ and }r\neq 0\\M\in \mathrm{R}\text{, }&\left(r=-\sqrt{2}h+h\text{ and }h\neq 0\right)\text{ or }\left(r=\sqrt{2}h+h\text{ and }h\neq 0\right)\text{ or }\left(m=0\text{ and }r\neq -h\text{ and }r\neq 0\right)\text{ or }\left(G=0\text{ and }r\neq -h\text{ and }r\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by r^{2}\left(r+h\right)^{2}, the least common multiple of \left(r+h\right)^{2},r^{2}.

$$2Gr^{2}Mm=G\left(r+h\right)^{2}Mm$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(r+h\right)^{2}.

$$2Gr^{2}Mm=G\left(r^{2}+2rh+h^{2}\right)Mm$$

Use the distributive property to multiply G by r^{2}+2rh+h^{2}.

$$2Gr^{2}Mm=\left(Gr^{2}+2Grh+Gh^{2}\right)Mm$$

Use the distributive property to multiply Gr^{2}+2Grh+Gh^{2} by M.

$$2Gr^{2}Mm=\left(Gr^{2}M+2GrhM+Gh^{2}M\right)m$$

Use the distributive property to multiply Gr^{2}M+2GrhM+Gh^{2}M by m.

$$2Gr^{2}Mm=Gr^{2}Mm+2GrhMm+Gh^{2}Mm$$

Subtract Gr^{2}Mm from both sides.

$$2Gr^{2}Mm-Gr^{2}Mm=2GrhMm+Gh^{2}Mm$$

Combine 2Gr^{2}Mm and -Gr^{2}Mm to get Gr^{2}Mm.

$$Gr^{2}Mm=2GrhMm+Gh^{2}Mm$$

Subtract 2GrhMm from both sides.

$$Gr^{2}Mm-2GrhMm=Gh^{2}Mm$$

Subtract Gh^{2}Mm from both sides.

$$Gr^{2}Mm-2GrhMm-Gh^{2}Mm=0$$

Reorder the terms.

$$GMmr^{2}-2GMhmr-GMmh^{2}=0$$

Combine all terms containing M.

$$\left(Gmr^{2}-2Ghmr-Gmh^{2}\right)M=0$$

Divide 0 by Gmr^{2}-2Ghmr-Gmh^{2}.

$$M=0$$