Solve frac{sqrt{x+7y}}{3x}=8012

$$\frac { \sqrt { x + 7 y } } { 3 x } = 8012$$

Solve for y

$$y=-\frac{x\left(1-577729296x\right)}{7}$$

$$x>0$$

Steps for Solving Radical Equations

Divide both sides by \frac{1}{3}x^{-1}.

$$\frac{\frac{1}{3x}\sqrt{7y+x}\times 3x}{1}=\frac{8012\times 3x}{1}$$

Dividing by \frac{1}{3}x^{-1} undoes the multiplication by \frac{1}{3}x^{-1}.

$$\sqrt{7y+x}=\frac{8012\times 3x}{1}$$

Divide 8012 by \frac{1}{3}x^{-1}.

$$\sqrt{7y+x}=24036x$$

Square both sides of the equation.

$$7y+x=577729296x^{2}$$

Subtract x from both sides of the equation.

$$7y+x-x=577729296x^{2}-x$$

Subtracting x from itself leaves 0.

$$7y=577729296x^{2}-x$$

Subtract x from 577729296x^{2}.

$$7y=x\left(577729296x-1\right)$$

Divide both sides by 7.

$$\frac{7y}{7}=\frac{x\left(577729296x-1\right)}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$y=\frac{x\left(577729296x-1\right)}{7}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{16176420288y+1}+1}{1155458592}\text{, }&arg(\frac{-\sqrt{16176420288y+1}+1}{1155458592})<\pi \text{ and }y\neq 0\\x=\frac{\sqrt{16176420288y+1}+1}{1155458592}\text{, }&arg(\frac{\sqrt{16176420288y+1}+1}{1155458592})<\pi \end{matrix}\right.$$

Solve for y (complex solution)

$$y=-\frac{x\left(1-577729296x\right)}{7}$$

$$arg(x)<\pi \text{ and }x\neq 0$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{16176420288y+1}+1}{1155458592}\text{, }&y\geq -\frac{1}{16176420288}\\x=\frac{-\sqrt{16176420288y+1}+1}{1155458592}\text{, }&y\geq -\frac{1}{16176420288}\text{ and }y<0\end{matrix}\right.$$