Solve M*(2a^{2}b^{2}+5ab^{2}-7ab)=-8a^{3}b^{4}-20a^2b^5+28a^2b^3

$$M \cdot ( 2 a ^ { 2 } b ^ { 2 } + 5 a b ^ { 2 } - 7 a b ) = - 8 a ^ { 3 } b ^ { 4 } - 20 a ^ { 2 } b ^ { 5 } + 28 a ^ { 2 } b ^ { 3 }$$

Solve for M

$$\left\{\begin{matrix}M=-\frac{4ab^{2}\left(2ab+5b^{2}-7\right)}{2ab+5b-7}\text{, }&b=0\text{ or }a\neq -\frac{5}{2}+\frac{7}{2b}\\M\in \mathrm{R}\text{, }&b=0\text{ or }\left(b=1\text{ and }a=1\right)\text{ or }a=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply M by 2a^{2}b^{2}+5ab^{2}-7ab.

$$2Ma^{2}b^{2}+5Mab^{2}-7Mab=-8a^{3}b^{4}-20a^{2}b^{5}+28a^{2}b^{3}$$

Combine all terms containing M.

$$\left(2a^{2}b^{2}+5ab^{2}-7ab\right)M=-8a^{3}b^{4}-20a^{2}b^{5}+28a^{2}b^{3}$$

The equation is in standard form.

$$\left(2a^{2}b^{2}+5ab^{2}-7ab\right)M=28a^{2}b^{3}-20a^{2}b^{5}-8a^{3}b^{4}$$

Divide both sides by 2a^{2}b^{2}+5ab^{2}-7ab.

$$\frac{\left(2a^{2}b^{2}+5ab^{2}-7ab\right)M}{2a^{2}b^{2}+5ab^{2}-7ab}=\frac{4a^{2}\left(7-5b^{2}-2ab\right)b^{3}}{2a^{2}b^{2}+5ab^{2}-7ab}$$

Dividing by 2a^{2}b^{2}+5ab^{2}-7ab undoes the multiplication by 2a^{2}b^{2}+5ab^{2}-7ab.

$$M=\frac{4a^{2}\left(7-5b^{2}-2ab\right)b^{3}}{2a^{2}b^{2}+5ab^{2}-7ab}$$

Divide 4\left(-2ab-5b^{2}+7\right)a^{2}b^{3} by 2a^{2}b^{2}+5ab^{2}-7ab.

$$M=\frac{4ab^{2}\left(7-5b^{2}-2ab\right)}{2ab+5b-7}$$

Solve for a

$$\left\{\begin{matrix}\\a=0\text{, }&\text{unconditionally}\\a=\frac{\sqrt{\left(Mb+10b^{4}\right)^{2}-280b^{6}-40Mb^{4}+28Mb^{3}+196b^{4}}}{8b^{3}}-\frac{5b}{4}-\frac{M}{8b^{2}}+\frac{7}{4b}\text{; }a=-\frac{\sqrt{\left(Mb+10b^{4}\right)^{2}-280b^{6}-40Mb^{4}+28Mb^{3}+196b^{4}}}{8b^{3}}-\frac{5b}{4}-\frac{M}{8b^{2}}+\frac{7}{4b}\text{, }&\left(b\neq 0\text{ and }M\leq -10b^{3}-\frac{\sqrt{-25600b^{9}+61440b^{8}-35840b^{7}}}{8b^{2}}+20b^{2}-14b\right)\text{ or }\left(b\neq 0\text{ and }M\geq -10b^{3}+\frac{\sqrt{-25600b^{9}+61440b^{8}-35840b^{7}}}{8b^{2}}+20b^{2}-14b\right)\text{ or }\left(b>0\text{ and }b\leq 1\right)\text{ or }b\geq \frac{7}{5}\\a\in \mathrm{R}\text{, }&b=0\end{matrix}\right.$$