Solve y=[(8/5x^4+3x)*(9/5X+sqrt{16x^14})]

$$y = [ ( \frac { 8 } { 5 } x ^ { 4 } + 3 x ) \cdot ( \frac { 9 } { 5 } X + \sqrt { 16 x ^ { 14 } } ) ]$$

Solve for X

$$\left\{\begin{matrix}X=-\frac{5\left(32x^{4}\left(|x|\right)^{7}+60x\left(|x|\right)^{7}-5y\right)}{9x\left(8x^{3}+15\right)}\text{, }&x\neq -\frac{\sqrt[3]{15}}{2}\text{ and }x\neq 0\\X\in \mathrm{R}\text{, }&\left(x=0\text{ or }x=-\frac{\sqrt[3]{15}}{2}\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{8}{5}x^{4}+3x by \frac{9}{5}X+\sqrt{16x^{14}}.

$$y=\frac{72}{25}x^{4}X+\frac{8}{5}x^{4}\sqrt{16x^{14}}+\frac{27}{5}xX+3x\sqrt{16x^{14}}$$

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

$$\frac{72}{25}x^{4}X+\frac{8}{5}x^{4}\sqrt{16x^{14}}+\frac{27}{5}xX+3x\sqrt{16x^{14}}=y$$

Subtract \frac{8}{5}x^{4}\sqrt{16x^{14}} from both sides.

$$\frac{72}{25}x^{4}X+\frac{27}{5}xX+3x\sqrt{16x^{14}}=y-\frac{8}{5}x^{4}\sqrt{16x^{14}}$$

Subtract 3x\sqrt{16x^{14}} from both sides.

$$\frac{72}{25}x^{4}X+\frac{27}{5}xX=y-\frac{8}{5}x^{4}\sqrt{16x^{14}}-3x\sqrt{16x^{14}}$$

Combine all terms containing X.

$$\left(\frac{72}{25}x^{4}+\frac{27}{5}x\right)X=y-\frac{8}{5}x^{4}\sqrt{16x^{14}}-3x\sqrt{16x^{14}}$$

The equation is in standard form.

$$\left(\frac{72x^{4}}{25}+\frac{27x}{5}\right)X=-\frac{8x^{4}\sqrt{16x^{14}}}{5}-3x\sqrt{16x^{14}}+y$$

Divide both sides by \frac{72}{25}x^{4}+\frac{27}{5}x.

$$\frac{\left(\frac{72x^{4}}{25}+\frac{27x}{5}\right)X}{\frac{72x^{4}}{25}+\frac{27x}{5}}=\frac{-\frac{32x^{4}\left(|x|\right)^{7}}{5}-12x\left(|x|\right)^{7}+y}{\frac{72x^{4}}{25}+\frac{27x}{5}}$$

Dividing by \frac{72}{25}x^{4}+\frac{27}{5}x undoes the multiplication by \frac{72}{25}x^{4}+\frac{27}{5}x.

$$X=\frac{-\frac{32x^{4}\left(|x|\right)^{7}}{5}-12x\left(|x|\right)^{7}+y}{\frac{72x^{4}}{25}+\frac{27x}{5}}$$

Divide y-\frac{32x^{4}\left(|x|\right)^{7}}{5}-12x\left(|x|\right)^{7} by \frac{72}{25}x^{4}+\frac{27}{5}x.

$$X=\frac{5\left(-32x^{4}\left(|x|\right)^{7}-60x\left(|x|\right)^{7}+5y\right)}{9x\left(8x^{3}+15\right)}$$