Solve 14x^7y^4:(-7xy)=-2x6y3

$$14 x ^ { 7 } y ^ { 4 } : ( - 7 x y ) = - 2 x 6 y 3$$

Solve for x_6

$$x_{6}=\frac{y^{3}x^{6}}{y_{3}}$$

$$y_{3}\neq 0\text{ and }y\neq 0\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by -7xy.

$$14x^{7}y^{4}=-2x_{6}y_{3}\left(-7\right)xy$$

Multiply -2 and -7 to get 14.

$$14x^{7}y^{4}=14x_{6}y_{3}xy$$

Swap sides so that all variable terms are on the left hand side.

$$14x_{6}y_{3}xy=14x^{7}y^{4}$$

Cancel out 14 on both sides.

$$x_{6}y_{3}xy=x^{7}y^{4}$$

The equation is in standard form.

$$xyy_{3}x_{6}=y^{4}x^{7}$$

Divide both sides by y_{3}xy.

$$\frac{xyy_{3}x_{6}}{xyy_{3}}=\frac{y^{4}x^{7}}{xyy_{3}}$$

Dividing by y_{3}xy undoes the multiplication by y_{3}xy.

$$x_{6}=\frac{y^{4}x^{7}}{xyy_{3}}$$

Divide x^{7}y^{4} by y_{3}xy.

$$x_{6}=\frac{y^{3}x^{6}}{y_{3}}$$

Solve for x (complex solution)

$$x\in e^{\frac{\pi i}{3}}y^{-\frac{1}{2}}\sqrt[6]{x_{6}}\sqrt[6]{y_{3}},y^{-\frac{1}{2}}\sqrt[6]{x_{6}}\sqrt[6]{y_{3}},e^{\frac{2\pi i}{3}}y^{-\frac{1}{2}}\sqrt[6]{x_{6}}\sqrt[6]{y_{3}},-y^{-\frac{1}{2}}\sqrt[6]{x_{6}}\sqrt[6]{y_{3}},e^{\frac{4\pi i}{3}}y^{-\frac{1}{2}}\sqrt[6]{x_{6}}\sqrt[6]{y_{3}},e^{\frac{5\pi i}{3}}y^{-\frac{1}{2}}\sqrt[6]{x_{6}}\sqrt[6]{y_{3}}$$

$$y\neq 0\text{ and }y_{3}\neq 0\text{ and }x_{6}\neq 0$$

Solve for x

$$x=\sqrt[6]{\frac{x_{6}y_{3}}{y^{3}}}$$

$$x=-\sqrt[6]{\frac{x_{6}y_{3}}{y^{3}}}\text{, }\left(y<0\text{ and }x_{6}<0\text{ and }y_{3}>0\right)\text{ or }\left(y>0\text{ and }x_{6}>0\text{ and }y_{3}>0\right)\text{ or }\left(y<0\text{ and }y_{3}<0\text{ and }x_{6}>0\right)\text{ or }\left(y>0\text{ and }x_{6}<0\text{ and }y_{3}<0\right)$$