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

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

Solve for x, y

$$x=\frac{12\sqrt{3}-4}{13}\approx 1.291123822\text{, }y=\frac{-3\sqrt{3}-12}{13}\approx -1.322780956$$

$$x=\frac{-12\sqrt{3}-4}{13}\approx -1.906508438\text{, }y=\frac{12-3\sqrt{3}}{13}\approx 0.523372891$$

Steps Using Substitution and Quadratic Formula

Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 4,3.

$$3x^{2}+4y^{2}=12$$

Consider the second equation. Express \left(-\frac{\sqrt{3}}{3}\right)x as a single fraction.

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

Since \frac{-\sqrt{3}x}{3} and \frac{\sqrt{3}}{3} have the same denominator, subtract them by subtracting their numerators.

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

Subtract \frac{-\sqrt{3}x-\sqrt{3}}{3} from both sides.

$$y-\frac{-\sqrt{3}x-\sqrt{3}}{3}=0$$

Multiply both sides of the equation by 3.

$$3y-\left(-\sqrt{3}x-\sqrt{3}\right)=0$$

To find the opposite of -\sqrt{3}x-\sqrt{3}, find the opposite of each term.

$$3y+\sqrt{3}x+\sqrt{3}=0$$

Subtract \sqrt{3} from both sides. Anything subtracted from zero gives its negation.

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

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.

$$3y+\sqrt{3}x=-\sqrt{3},3x^{2}+4y^{2}=12$$

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

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

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

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

Divide both sides by 3.

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

Substitute \left(-\frac{\sqrt{3}}{3}\right)x-\frac{\sqrt{3}}{3} for y in the other equation, 3x^{2}+4y^{2}=12.

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

Square \left(-\frac{\sqrt{3}}{3}\right)x-\frac{\sqrt{3}}{3}.

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

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

$$3x^{2}+4\left(-\frac{\sqrt{3}}{3}\right)^{2}x^{2}+8\left(-\frac{\sqrt{3}}{3}\right)^{2}x+4\left(-\frac{\sqrt{3}}{3}\right)^{2}=12$$

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

$$\left(4\left(-\frac{\sqrt{3}}{3}\right)^{2}+3\right)x^{2}+8\left(-\frac{\sqrt{3}}{3}\right)^{2}x+4\left(-\frac{\sqrt{3}}{3}\right)^{2}=12$$

Subtract 12 from both sides of the equation.

$$\left(4\left(-\frac{\sqrt{3}}{3}\right)^{2}+3\right)x^{2}+8\left(-\frac{\sqrt{3}}{3}\right)^{2}x+4\left(-\frac{\sqrt{3}}{3}\right)^{2}-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3+4\left(-\frac{\sqrt{3}}{3}\right)^{2} for a, 4\times 2\left(-\frac{\sqrt{3}}{3}\right)\left(-\frac{\sqrt{3}}{3}\right) for b, and -\frac{32}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8\left(-\frac{\sqrt{3}}{3}\right)^{2}±\sqrt{\left(8\left(-\frac{\sqrt{3}}{3}\right)^{2}\right)^{2}-4\left(4\left(-\frac{\sqrt{3}}{3}\right)^{2}+3\right)\left(-\frac{32}{3}\right)}}{2\left(4\left(-\frac{\sqrt{3}}{3}\right)^{2}+3\right)}$$

Square 4\times 2\left(-\frac{\sqrt{3}}{3}\right)\left(-\frac{\sqrt{3}}{3}\right).

$$x=\frac{-8\left(-\frac{\sqrt{3}}{3}\right)^{2}±\sqrt{\frac{64}{9}-4\left(4\left(-\frac{\sqrt{3}}{3}\right)^{2}+3\right)\left(-\frac{32}{3}\right)}}{2\left(4\left(-\frac{\sqrt{3}}{3}\right)^{2}+3\right)}$$

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

$$x=\frac{-8\left(-\frac{\sqrt{3}}{3}\right)^{2}±\sqrt{\frac{64}{9}-\frac{52}{3}\left(-\frac{32}{3}\right)}}{2\left(4\left(-\frac{\sqrt{3}}{3}\right)^{2}+3\right)}$$

Multiply -\frac{52}{3} times -\frac{32}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-8\left(-\frac{\sqrt{3}}{3}\right)^{2}±\sqrt{\frac{64+1664}{9}}}{2\left(4\left(-\frac{\sqrt{3}}{3}\right)^{2}+3\right)}$$

Add \frac{64}{9} to \frac{1664}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-8\left(-\frac{\sqrt{3}}{3}\right)^{2}±\sqrt{192}}{2\left(4\left(-\frac{\sqrt{3}}{3}\right)^{2}+3\right)}$$

Take the square root of 192.

$$x=\frac{-8\left(-\frac{\sqrt{3}}{3}\right)^{2}±8\sqrt{3}}{2\left(4\left(-\frac{\sqrt{3}}{3}\right)^{2}+3\right)}$$

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

$$x=\frac{-\frac{8}{3}±8\sqrt{3}}{\frac{26}{3}}$$

Now solve the equation x=\frac{-\frac{8}{3}±8\sqrt{3}}{\frac{26}{3}} when ± is plus. Add -\frac{8}{3} to 8\sqrt{3}.

$$x=\frac{8\sqrt{3}-\frac{8}{3}}{\frac{26}{3}}$$

Divide 8\sqrt{3}-\frac{8}{3} by \frac{26}{3} by multiplying 8\sqrt{3}-\frac{8}{3} by the reciprocal of \frac{26}{3}.

$$x=\frac{12\sqrt{3}-4}{13}$$

Now solve the equation x=\frac{-\frac{8}{3}±8\sqrt{3}}{\frac{26}{3}} when ± is minus. Subtract 8\sqrt{3} from -\frac{8}{3}.

$$x=\frac{-8\sqrt{3}-\frac{8}{3}}{\frac{26}{3}}$$

Divide -\frac{8}{3}-8\sqrt{3} by \frac{26}{3} by multiplying -\frac{8}{3}-8\sqrt{3} by the reciprocal of \frac{26}{3}.

$$x=\frac{-12\sqrt{3}-4}{13}$$

There are two solutions for x: \frac{12\sqrt{3}-4}{13} and \frac{-4-12\sqrt{3}}{13}. Substitute \frac{12\sqrt{3}-4}{13} for x in the equation y=\left(-\frac{\sqrt{3}}{3}\right)x-\frac{\sqrt{3}}{3} to find the corresponding solution for y that satisfies both equations.

$$y=\left(-\frac{\sqrt{3}}{3}\right)\times \frac{12\sqrt{3}-4}{13}-\frac{\sqrt{3}}{3}$$

Now substitute \frac{-4-12\sqrt{3}}{13} for x in the equation y=\left(-\frac{\sqrt{3}}{3}\right)x-\frac{\sqrt{3}}{3} and solve to find the corresponding solution for y that satisfies both equations.

$$y=\left(-\frac{\sqrt{3}}{3}\right)\times \frac{-12\sqrt{3}-4}{13}-\frac{\sqrt{3}}{3}$$

The system is now solved.

$$y=\left(-\frac{\sqrt{3}}{3}\right)\times \frac{12\sqrt{3}-4}{13}-\frac{\sqrt{3}}{3},x=\frac{12\sqrt{3}-4}{13}\text{ or }y=\left(-\frac{\sqrt{3}}{3}\right)\times \frac{-12\sqrt{3}-4}{13}-\frac{\sqrt{3}}{3},x=\frac{-12\sqrt{3}-4}{13}$$