Solve y^2(a+x)=x^2(3a-x)

$$y ^ { 2 } ( a + x ) = x ^ { 2 } ( 3 a - x )$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{x\left(x^{2}+y^{2}\right)}{y^{2}-3x^{2}}\text{, }&|y|\neq \sqrt{3}|x|\\a\in \mathrm{R}\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply y^{2} by a+x.

$$y^{2}a+y^{2}x=x^{2}\left(3a-x\right)$$

Use the distributive property to multiply x^{2} by 3a-x.

$$y^{2}a+y^{2}x=3x^{2}a-x^{3}$$

Subtract 3x^{2}a from both sides.

$$y^{2}a+y^{2}x-3x^{2}a=-x^{3}$$

Subtract y^{2}x from both sides.

$$y^{2}a-3x^{2}a=-x^{3}-y^{2}x$$

Combine all terms containing a.

$$\left(y^{2}-3x^{2}\right)a=-x^{3}-y^{2}x$$

The equation is in standard form.

$$\left(y^{2}-3x^{2}\right)a=-x^{3}-xy^{2}$$

Divide both sides by y^{2}-3x^{2}.

$$\frac{\left(y^{2}-3x^{2}\right)a}{y^{2}-3x^{2}}=-\frac{x\left(x^{2}+y^{2}\right)}{y^{2}-3x^{2}}$$

Dividing by y^{2}-3x^{2} undoes the multiplication by y^{2}-3x^{2}.

$$a=-\frac{x\left(x^{2}+y^{2}\right)}{y^{2}-3x^{2}}$$