Solve a/x/y=a/2+frac{a/x}{frac{sqrt{3}}{4}a}

$$\frac{ \frac{ a }{ x } }{ y } = \frac{ \frac{ a }{ 2 } + \frac{ a }{ x } }{ \frac{ \sqrt{ 3 } }{ 4 } a }$$

Solve for x

$$x=\frac{\sqrt{3}a}{2y}-2$$

$$a\neq \frac{4\sqrt{3}y}{3}\text{ and }y\neq 0\text{ and }a\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by y.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and x is 2x. Multiply \frac{a}{2} times \frac{x}{x}. Multiply \frac{a}{x} times \frac{2}{2}.

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

Since \frac{ax}{2x} and \frac{2a}{2x} have the same denominator, add them by adding their numerators.

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

Multiply \frac{4}{3} times \frac{ax+2a}{2x} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2 in both numerator and denominator.

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

Use the distributive property to multiply 2 by ax+2a.

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

Express \frac{2ax+4a}{3x}y as a single fraction.

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

Express \frac{\left(2ax+4a\right)y}{3x}a^{-1} as a single fraction.

$$\frac{a}{x}=\frac{\left(2ax+4a\right)ya^{-1}}{3x}\times 3^{\frac{1}{2}}$$

Subtract \frac{\left(2ax+4a\right)ya^{-1}}{3x}\times 3^{\frac{1}{2}} from both sides.

$$\frac{a}{x}-\frac{\left(2ax+4a\right)ya^{-1}}{3x}\times 3^{\frac{1}{2}}=0$$

Use the distributive property to multiply 2ax+4a by y.

$$\frac{a}{x}-\frac{\left(2axy+4ay\right)a^{-1}}{3x}\times 3^{\frac{1}{2}}=0$$

Use the distributive property to multiply 2axy+4ay by a^{-1}.

$$\frac{a}{x}-\frac{2axya^{-1}+4aya^{-1}}{3x}\times 3^{\frac{1}{2}}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x, the least common multiple of x,3x.

$$3a-\left(2axya^{-1}+4aya^{-1}\right)\times 3^{\frac{1}{2}}=0$$

Reorder the terms.

$$3a-\sqrt{3}\left(2\times \frac{1}{a}axy+4\times \frac{1}{a}ay\right)=0$$

Multiply both sides of the equation by a.

$$3aa-\sqrt{3}\left(2\times \frac{1}{a}axy+4\times \frac{1}{a}ay\right)a=0$$

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

$$3a^{2}-\sqrt{3}\left(2\times \frac{1}{a}axy+4\times \frac{1}{a}ay\right)a=0$$

Express 2\times \frac{1}{a} as a single fraction.

$$3a^{2}-\sqrt{3}\left(\frac{2}{a}axy+4\times \frac{1}{a}ay\right)a=0$$

Express \frac{2}{a}a as a single fraction.

$$3a^{2}-\sqrt{3}\left(\frac{2a}{a}xy+4\times \frac{1}{a}ay\right)a=0$$

Cancel out a in both numerator and denominator.

$$3a^{2}-\sqrt{3}\left(2xy+4\times \frac{1}{a}ay\right)a=0$$

Express 4\times \frac{1}{a} as a single fraction.

$$3a^{2}-\sqrt{3}\left(2xy+\frac{4}{a}ay\right)a=0$$

Express \frac{4}{a}a as a single fraction.

$$3a^{2}-\sqrt{3}\left(2xy+\frac{4a}{a}y\right)a=0$$

Cancel out a in both numerator and denominator.

$$3a^{2}-\sqrt{3}\left(2xy+4y\right)a=0$$

Use the distributive property to multiply -\sqrt{3} by 2xy+4y.

$$3a^{2}+\left(-2\sqrt{3}xy-4\sqrt{3}y\right)a=0$$

Use the distributive property to multiply -2\sqrt{3}xy-4\sqrt{3}y by a.

$$3a^{2}-2\sqrt{3}xya-4\sqrt{3}ya=0$$

Subtract 3a^{2} from both sides. Anything subtracted from zero gives its negation.

$$-2\sqrt{3}xya-4\sqrt{3}ya=-3a^{2}$$

Add 4\sqrt{3}ya to both sides.

$$-2\sqrt{3}xya=-3a^{2}+4\sqrt{3}ya$$

The equation is in standard form.

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

Divide both sides by -2\sqrt{3}ya.

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

Dividing by -2\sqrt{3}ya undoes the multiplication by -2\sqrt{3}ya.

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

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

$$x=\frac{\sqrt{3}a}{2y}-2$$

Variable x cannot be equal to 0.

$$x=\frac{\sqrt{3}a}{2y}-2\text{, }x\neq 0$$

Solve for a

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

$$x\neq -2\text{ and }y\neq 0\text{ and }x\neq 0$$