Solve x-4/y=x-3/sqrt{{left(frac{2{3}xright)}^2+{left(frac{2sqrt{2}}{3}xright)}^2}}

$$\frac{ x-4 }{ y } = \frac{ x-3 }{ \sqrt{ { \left( \frac{ 2 }{ 3 } x \right) }^{ 2 } + { \left( \frac{ 2 \sqrt{ 2 } }{ 3 } x \right) }^{ 2 } } }$$

Solve for y (complex solution)

$$y=-\frac{2\sqrt{3x^{2}}\left(x-4\right)}{3\left(3-x\right)}$$

$$x\neq 0\text{ and }x\neq 4\text{ and }x\neq 3$$

Steps for Solving Linear Equation

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by y.

$$x-4=\frac{3}{2}y\times \left(3x^{2}\right)^{-\frac{1}{2}}\left(x-3\right)$$

Expand \left(3x^{2}\right)^{-\frac{1}{2}}.

$$x-4=\frac{3}{2}y\times 3^{-\frac{1}{2}}\left(x^{2}\right)^{-\frac{1}{2}}\left(x-3\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and -\frac{1}{2} to get -1.

$$x-4=\frac{3}{2}y\times 3^{-\frac{1}{2}}x^{-1}\left(x-3\right)$$

Use the distributive property to multiply \frac{3}{2}y\times 3^{-\frac{1}{2}}x^{-1} by x-3.

$$x-4=\frac{3}{2}y\times 3^{-\frac{1}{2}}x^{-1}x-\frac{3}{2}y\times 3^{\frac{1}{2}}x^{-1}$$

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

$$\frac{3}{2}y\times 3^{-\frac{1}{2}}x^{-1}x-\frac{3}{2}y\times 3^{\frac{1}{2}}x^{-1}=x-4$$

Reorder the terms.

$$\frac{3}{2}\times 3^{-\frac{1}{2}}\times \frac{1}{x}xy-\frac{3}{2}\sqrt{3}\times \frac{1}{x}y=x-4$$

Multiply both sides of the equation by 2x, the least common multiple of 2,x.

$$\frac{3}{2}\times 3^{-\frac{1}{2}}\times 2\times 1xy-\frac{3}{2}\sqrt{3}\times 2y=2xx+2x\left(-4\right)$$

Multiply \frac{3}{2} and 2 to get 3.

$$3\times 3^{-\frac{1}{2}}\times 1xy-\frac{3}{2}\sqrt{3}\times 2y=2xx+2x\left(-4\right)$$

To multiply powers of the same base, add their exponents. Add 1 and -\frac{1}{2} to get \frac{1}{2}.

$$3^{\frac{1}{2}}\times 1xy-\frac{3}{2}\sqrt{3}\times 2y=2xx+2x\left(-4\right)$$

Multiply -\frac{3}{2} and 2 to get -3.

$$3^{\frac{1}{2}}\times 1xy-3\sqrt{3}y=2xx+2x\left(-4\right)$$

Multiply x and x to get x^{2}.

$$3^{\frac{1}{2}}\times 1xy-3\sqrt{3}y=2x^{2}+2x\left(-4\right)$$

Multiply 2 and -4 to get -8.

$$3^{\frac{1}{2}}\times 1xy-3\sqrt{3}y=2x^{2}-8x$$

Reorder the terms.

$$\sqrt{3}xy-3\sqrt{3}y=2x^{2}-8x$$

Combine all terms containing y.

$$\left(\sqrt{3}x-3\sqrt{3}\right)y=2x^{2}-8x$$

Divide both sides by \sqrt{3}x-3\sqrt{3}.

$$\frac{\left(\sqrt{3}x-3\sqrt{3}\right)y}{\sqrt{3}x-3\sqrt{3}}=\frac{2x\left(x-4\right)}{\sqrt{3}x-3\sqrt{3}}$$

Dividing by \sqrt{3}x-3\sqrt{3} undoes the multiplication by \sqrt{3}x-3\sqrt{3}.

$$y=\frac{2x\left(x-4\right)}{\sqrt{3}x-3\sqrt{3}}$$

Divide 2x\left(-4+x\right) by \sqrt{3}x-3\sqrt{3}.

$$y=\frac{2\sqrt{3}x\left(x-4\right)}{3\left(x-3\right)}$$

Variable y cannot be equal to 0.

$$y=\frac{2\sqrt{3}x\left(x-4\right)}{3\left(x-3\right)}\text{, }y\neq 0$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{3y^{2}+8\sqrt{3}y+64}-\sqrt{3}y+8}{4}\text{; }x=\frac{-\sqrt{3y^{2}-8\sqrt{3}y+64}+\sqrt{3}y+8}{4}\text{, }&y>0\\x=\frac{\sqrt{3y^{2}-8\sqrt{3}y+64}+\sqrt{3}y+8}{4}\text{, }&y\neq 0\end{matrix}\right.$$

Solve for y

$$y=-\frac{2\sqrt{3}|x|\left(x-4\right)}{3\left(3-x\right)}$$

$$x\neq 4\text{ and }x\neq 0\text{ and }x\neq 3$$