Solve operatorname{ek}frac{2+sqrt{3}}{a+1}=1/2-sqrt{3}

$$\operatorname { ek } \frac { 2 + \sqrt { 3 } } { a + 1 } = \frac { 1 } { 2 - \sqrt { 3 } }$$

Solve for a

$$a=ek-1$$

$$k\neq 0$$

Steps for Solving Linear Equation

Variable a cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by a+1.

$$ek\left(2+\sqrt{3}\right)=\left(a+1\right)\left(3^{\frac{1}{2}}+2\right)$$

Use the distributive property to multiply ek by 2+\sqrt{3}.

$$2ek+ek\sqrt{3}=\left(a+1\right)\left(3^{\frac{1}{2}}+2\right)$$

Use the distributive property to multiply a+1 by 3^{\frac{1}{2}}+2.

$$2ek+ek\sqrt{3}=a\times 3^{\frac{1}{2}}+2a+3^{\frac{1}{2}}+2$$

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

$$a\times 3^{\frac{1}{2}}+2a+3^{\frac{1}{2}}+2=2ek+ek\sqrt{3}$$

Subtract 3^{\frac{1}{2}} from both sides.

$$a\times 3^{\frac{1}{2}}+2a+2=2ek+ek\sqrt{3}-3^{\frac{1}{2}}$$

Subtract 2 from both sides.

$$a\times 3^{\frac{1}{2}}+2a=2ek+ek\sqrt{3}-3^{\frac{1}{2}}-2$$

Reorder the terms.

$$\sqrt{3}a+2a=e\sqrt{3}k+2ek-\sqrt{3}-2$$

Combine all terms containing a.

$$\left(\sqrt{3}+2\right)a=e\sqrt{3}k+2ek-\sqrt{3}-2$$

Divide both sides by 2+\sqrt{3}.

$$\frac{\left(\sqrt{3}+2\right)a}{\sqrt{3}+2}=\frac{e\sqrt{3}k+2ek-\sqrt{3}-2}{\sqrt{3}+2}$$

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

$$a=\frac{e\sqrt{3}k+2ek-\sqrt{3}-2}{\sqrt{3}+2}$$

Divide e\sqrt{3}k+2ek-\sqrt{3}-2 by 2+\sqrt{3}.

$$a=ek-1$$

Variable a cannot be equal to -1.

$$a=ek-1\text{, }a\neq -1$$

Solve for k

$$k=\frac{a+1}{e}$$

$$a\neq -1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by a+1.

$$ek\left(2+\sqrt{3}\right)=\left(a+1\right)\left(3^{\frac{1}{2}}+2\right)$$

Use the distributive property to multiply ek by 2+\sqrt{3}.

$$2ek+ek\sqrt{3}=\left(a+1\right)\left(3^{\frac{1}{2}}+2\right)$$

Use the distributive property to multiply a+1 by 3^{\frac{1}{2}}+2.

$$2ek+ek\sqrt{3}=a\times 3^{\frac{1}{2}}+2a+3^{\frac{1}{2}}+2$$

Reorder the terms.

$$e\sqrt{3}k+2ek=\sqrt{3}a+2a+\sqrt{3}+2$$

Combine all terms containing k.

$$\left(e\sqrt{3}+2e\right)k=\sqrt{3}a+2a+\sqrt{3}+2$$

Divide both sides by e\sqrt{3}+2e.

$$\frac{\left(e\sqrt{3}+2e\right)k}{e\sqrt{3}+2e}=\frac{\sqrt{3}a+2a+\sqrt{3}+2}{e\sqrt{3}+2e}$$

Dividing by e\sqrt{3}+2e undoes the multiplication by e\sqrt{3}+2e.

$$k=\frac{\sqrt{3}a+2a+\sqrt{3}+2}{e\sqrt{3}+2e}$$

Divide \sqrt{3}a+2a+\sqrt{3}+2 by e\sqrt{3}+2e.

$$k=\frac{a+1}{e}$$