Solve {l}{y=sqrt{3}x-frac{sqrt{3}}{2}}{x^2+y^2/4=1}

$$\left. \begin{array} { l } { y = \sqrt { 3 } x - \frac { \sqrt { 3 } } { 2 } } \\ { x ^ { 2 } + \frac { y ^ { 2 } } { 4 } = 1 } \end{array} \right.$$

Solve for y, x

$$x=-\frac{1}{2}=-0.5\text{, }y=-\sqrt{3}\approx -1.732050808$$

$$x=\frac{13}{14}\approx 0.928571429\text{, }y=\frac{3\sqrt{3}}{7}\approx 0.742307489$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Subtract \sqrt{3}x from both sides.

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

Multiply both sides of the equation by 2.

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

Consider the second equation. Multiply both sides of the equation by 4.

$$4x^{2}+y^{2}=4$$

To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.

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

Solve 2y+\left(-2\sqrt{3}\right)x=-\sqrt{3} for y by isolating y on the left hand side of the equal sign.

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

Subtract \left(-2\sqrt{3}\right)x from both sides of the equation.

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

Divide both sides by 2.

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

Substitute \sqrt{3}x-\frac{\sqrt{3}}{2} for y in the other equation, 4x^{2}+y^{2}=4.

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

Square \sqrt{3}x-\frac{\sqrt{3}}{2}.

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

Add 4x^{2} to \left(\sqrt{3}\right)^{2}x^{2}.

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

Subtract 4 from both sides of the equation.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4+1\left(\sqrt{3}\right)^{2} for a, 1\times 2\sqrt{3}\left(-\frac{\sqrt{3}}{2}\right) for b, and -\frac{13}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2\sqrt{3}\left(-\frac{\sqrt{3}}{2}\right)±\sqrt{\left(2\sqrt{3}\left(-\frac{\sqrt{3}}{2}\right)\right)^{2}-4\left(\left(\sqrt{3}\right)^{2}+4\right)\left(-\frac{13}{4}\right)}}{2\left(\left(\sqrt{3}\right)^{2}+4\right)}$$

Square 1\times 2\sqrt{3}\left(-\frac{\sqrt{3}}{2}\right).

$$x=\frac{-2\sqrt{3}\left(-\frac{\sqrt{3}}{2}\right)±\sqrt{9-4\left(\left(\sqrt{3}\right)^{2}+4\right)\left(-\frac{13}{4}\right)}}{2\left(\left(\sqrt{3}\right)^{2}+4\right)}$$

Multiply -4 times 4+1\left(\sqrt{3}\right)^{2}.

$$x=\frac{-2\sqrt{3}\left(-\frac{\sqrt{3}}{2}\right)±\sqrt{9-28\left(-\frac{13}{4}\right)}}{2\left(\left(\sqrt{3}\right)^{2}+4\right)}$$

Multiply -28 times -\frac{13}{4}.

$$x=\frac{-2\sqrt{3}\left(-\frac{\sqrt{3}}{2}\right)±\sqrt{9+91}}{2\left(\left(\sqrt{3}\right)^{2}+4\right)}$$

Add 9 to 91.

$$x=\frac{-2\sqrt{3}\left(-\frac{\sqrt{3}}{2}\right)±\sqrt{100}}{2\left(\left(\sqrt{3}\right)^{2}+4\right)}$$

Take the square root of 100.

$$x=\frac{-2\sqrt{3}\left(-\frac{\sqrt{3}}{2}\right)±10}{2\left(\left(\sqrt{3}\right)^{2}+4\right)}$$

The opposite of 1\times 2\sqrt{3}\left(-\frac{\sqrt{3}}{2}\right) is 3.

$$x=\frac{3±10}{2\left(\left(\sqrt{3}\right)^{2}+4\right)}$$

Multiply 2 times 4+1\left(\sqrt{3}\right)^{2}.

$$x=\frac{3±10}{14}$$

Now solve the equation x=\frac{3±10}{14} when ± is plus. Add 3 to 10.

$$x=\frac{13}{14}$$

Now solve the equation x=\frac{3±10}{14} when ± is minus. Subtract 10 from 3.

$$x=-\frac{7}{14}$$

Reduce the fraction \frac{-7}{14} to lowest terms by extracting and canceling out 7.

$$x=-\frac{1}{2}$$

There are two solutions for x: \frac{13}{14} and -\frac{1}{2}. Substitute \frac{13}{14} for x in the equation y=\sqrt{3}x-\frac{\sqrt{3}}{2} to find the corresponding solution for y that satisfies both equations.

$$y=\sqrt{3}\times \frac{13}{14}-\frac{\sqrt{3}}{2}$$

Multiply \sqrt{3} times \frac{13}{14}.

$$y=\frac{13\sqrt{3}}{14}-\frac{\sqrt{3}}{2}$$

Now substitute -\frac{1}{2} for x in the equation y=\sqrt{3}x-\frac{\sqrt{3}}{2} and solve to find the corresponding solution for y that satisfies both equations.

$$y=\sqrt{3}\left(-\frac{1}{2}\right)-\frac{\sqrt{3}}{2}$$

Multiply \sqrt{3} times -\frac{1}{2}.

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

The system is now solved.

$$y=\frac{13\sqrt{3}}{14}-\frac{\sqrt{3}}{2},x=\frac{13}{14}\text{ or }y=\frac{-\sqrt{3}-\sqrt{3}}{2},x=-\frac{1}{2}$$