Solve {l}{x=sqrt{frac{(4+sqrt{3})(3-sqrt{3})}{4}}}{y=1/x}

$$\left. \begin{array} { l } { x = \sqrt { \frac { ( 4 + \sqrt { 3 } ) ( 3 - \sqrt { 3 } ) } { 4 } } } \\ { y = \frac { 1 } { x } } \end{array} \right.$$

Solve for x, y

$$x = \frac{\sqrt{9 - \sqrt{3}}}{2} \approx 1.347956712$$

$$y=\frac{\sqrt{78\left(\sqrt{3}+9\right)}}{39}\approx 0.741863586$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 4+\sqrt{3} by 3-\sqrt{3} and combine like terms.

$$x=\sqrt{\frac{12-\sqrt{3}-\left(\sqrt{3}\right)^{2}}{4}}$$

The square of \sqrt{3} is 3.

$$x=\sqrt{\frac{12-\sqrt{3}-3}{4}}$$

Subtract 3 from 12 to get 9.

$$x=\sqrt{\frac{9-\sqrt{3}}{4}}$$

Divide each term of 9-\sqrt{3} by 4 to get \frac{9}{4}-\frac{1}{4}\sqrt{3}.

$$x=\sqrt{\frac{9}{4}-\frac{1}{4}\sqrt{3}}$$

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

$$yx=1$$

Pick one of the two equations which is more simple to solve for x by isolating x on the left hand side of the equal sign.

$$x=\sqrt{\frac{9-\sqrt{3}}{4}}$$

Divide both sides by 1.

$$x=\frac{\sqrt{9-\sqrt{3}}}{2}$$

Substitute \frac{\sqrt{9-\sqrt{3}}}{2} for x in xy=1. Because the resulting equation contains only one variable, you can solve for y directly.

$$\frac{\sqrt{9-\sqrt{3}}}{2}y=1$$

Divide 1 by \frac{\sqrt{9-\sqrt{3}}}{2}.

$$y=\frac{\sqrt{78\sqrt{3}+702}}{39}$$

The system is now solved.

$$x=\frac{\sqrt{9-\sqrt{3}}}{2},y=\frac{\sqrt{78\sqrt{3}+702}}{39}$$