Solve 3ax^2-5y=6a-x

$$3 a x ^ { 2 } - 5 y = 6 a - x$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{x-5y}{3\left(x^{2}-2\right)}\text{, }&x\neq -\sqrt{2}\text{ and }x\neq \sqrt{2}\\a\in \mathrm{C}\text{, }&\left(x=\sqrt{2}\text{ and }y=\frac{\sqrt{2}}{5}\right)\text{ or }\left(x=-\sqrt{2}\text{ and }y=-\frac{\sqrt{2}}{5}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 6a from both sides.

$$3ax^{2}-5y-6a=-x$$

Add 5y to both sides.

$$3ax^{2}-6a=-x+5y$$

Combine all terms containing a.

$$\left(3x^{2}-6\right)a=-x+5y$$

The equation is in standard form.

$$\left(3x^{2}-6\right)a=5y-x$$

Divide both sides by 3x^{2}-6.

$$\frac{\left(3x^{2}-6\right)a}{3x^{2}-6}=\frac{5y-x}{3x^{2}-6}$$

Dividing by 3x^{2}-6 undoes the multiplication by 3x^{2}-6.

$$a=\frac{5y-x}{3x^{2}-6}$$

Divide -x+5y by 3x^{2}-6.

$$a=\frac{5y-x}{3\left(x^{2}-2\right)}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{x-5y}{3\left(x^{2}-2\right)}\text{, }&|x|\neq \sqrt{2}\\a\in \mathrm{R}\text{, }&\left(x=-\sqrt{2}\text{ and }y=-\frac{\sqrt{2}}{5}\right)\text{ or }\left(x=\sqrt{2}\text{ and }y=\frac{\sqrt{2}}{5}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 6a from both sides.

$$3ax^{2}-5y-6a=-x$$

Add 5y to both sides.

$$3ax^{2}-6a=-x+5y$$

Combine all terms containing a.

$$\left(3x^{2}-6\right)a=-x+5y$$

The equation is in standard form.

$$\left(3x^{2}-6\right)a=5y-x$$

Divide both sides by 3x^{2}-6.

$$\frac{\left(3x^{2}-6\right)a}{3x^{2}-6}=\frac{5y-x}{3x^{2}-6}$$

Dividing by 3x^{2}-6 undoes the multiplication by 3x^{2}-6.

$$a=\frac{5y-x}{3x^{2}-6}$$

Divide -x+5y by 3x^{2}-6.

$$a=\frac{5y-x}{3\left(x^{2}-2\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{60ay+72a^{2}+1}-1}{6a}\text{; }x=-\frac{\sqrt{60ay+72a^{2}+1}+1}{6a}\text{, }&a\neq 0\\x=5y\text{, }&a=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{60ay+72a^{2}+1}-1}{6a}\text{; }x=-\frac{\sqrt{60ay+72a^{2}+1}+1}{6a}\text{, }&a\neq 0\text{ and }\left(a\leq \frac{-\sqrt{25y^{2}-2}-5y}{12}\text{ or }|y|\leq \frac{\sqrt{2}}{5}\text{ or }a\geq \frac{\sqrt{25y^{2}-2}-5y}{12}\right)\\x=5y\text{, }&a=0\end{matrix}\right.$$