Solve {l}{x^2-4y^2=9}{5x+6y=7}

$$\left. \begin{array} { l } { x ^ { 2 } - 4 y ^ { 2 } = 9 } \\ { 5 x + 6 y = 7 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=\frac{35+3\sqrt{95}i}{16}\approx 2.1875+1.82752394i\text{, }y=\frac{-5\sqrt{95}i-21}{32}\approx -0.65625-1.522936616i$$

$$x=\frac{-3\sqrt{95}i+35}{16}\approx 2.1875-1.82752394i\text{, }y=\frac{-21+5\sqrt{95}i}{32}\approx -0.65625+1.522936616i$$

Steps Using Substitution and Quadratic Formula

Solve 5x+6y=7 for x by isolating x on the left hand side of the equal sign.

$$5x+6y=7$$

Subtract 6y from both sides of the equation.

$$5x=-6y+7$$

Divide both sides by 5.

$$x=-\frac{6}{5}y+\frac{7}{5}$$

Substitute -\frac{6}{5}y+\frac{7}{5} for x in the other equation, -4y^{2}+x^{2}=9.

$$-4y^{2}+\left(-\frac{6}{5}y+\frac{7}{5}\right)^{2}=9$$

Square -\frac{6}{5}y+\frac{7}{5}.

$$-4y^{2}+\frac{36}{25}y^{2}-\frac{84}{25}y+\frac{49}{25}=9$$

Add -4y^{2} to \frac{36}{25}y^{2}.

$$-\frac{64}{25}y^{2}-\frac{84}{25}y+\frac{49}{25}=9$$

Subtract 9 from both sides of the equation.

$$-\frac{64}{25}y^{2}-\frac{84}{25}y-\frac{176}{25}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4+1\left(-\frac{6}{5}\right)^{2} for a, 1\times \frac{7}{5}\left(-\frac{6}{5}\right)\times 2 for b, and -\frac{176}{25} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-\frac{84}{25}\right)±\sqrt{\left(-\frac{84}{25}\right)^{2}-4\left(-\frac{64}{25}\right)\left(-\frac{176}{25}\right)}}{2\left(-\frac{64}{25}\right)}$$

Square 1\times \frac{7}{5}\left(-\frac{6}{5}\right)\times 2.

$$y=\frac{-\left(-\frac{84}{25}\right)±\sqrt{\frac{7056}{625}-4\left(-\frac{64}{25}\right)\left(-\frac{176}{25}\right)}}{2\left(-\frac{64}{25}\right)}$$

Multiply -4 times -4+1\left(-\frac{6}{5}\right)^{2}.

$$y=\frac{-\left(-\frac{84}{25}\right)±\sqrt{\frac{7056}{625}+\frac{256}{25}\left(-\frac{176}{25}\right)}}{2\left(-\frac{64}{25}\right)}$$

Multiply \frac{256}{25} times -\frac{176}{25} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{84}{25}\right)±\sqrt{\frac{7056-45056}{625}}}{2\left(-\frac{64}{25}\right)}$$

Add \frac{7056}{625} to -\frac{45056}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{84}{25}\right)±\sqrt{-\frac{304}{5}}}{2\left(-\frac{64}{25}\right)}$$

Take the square root of -\frac{304}{5}.

$$y=\frac{-\left(-\frac{84}{25}\right)±\frac{4\sqrt{95}i}{5}}{2\left(-\frac{64}{25}\right)}$$

The opposite of 1\times \frac{7}{5}\left(-\frac{6}{5}\right)\times 2 is \frac{84}{25}.

$$y=\frac{\frac{84}{25}±\frac{4\sqrt{95}i}{5}}{2\left(-\frac{64}{25}\right)}$$

Multiply 2 times -4+1\left(-\frac{6}{5}\right)^{2}.

$$y=\frac{\frac{84}{25}±\frac{4\sqrt{95}i}{5}}{-\frac{128}{25}}$$

Now solve the equation y=\frac{\frac{84}{25}±\frac{4\sqrt{95}i}{5}}{-\frac{128}{25}} when ± is plus. Add \frac{84}{25} to \frac{4i\sqrt{95}}{5}.

$$y=\frac{\frac{4\sqrt{95}i}{5}+\frac{84}{25}}{-\frac{128}{25}}$$

Divide \frac{84}{25}+\frac{4i\sqrt{95}}{5} by -\frac{128}{25} by multiplying \frac{84}{25}+\frac{4i\sqrt{95}}{5} by the reciprocal of -\frac{128}{25}.

$$y=\frac{-5\sqrt{95}i-21}{32}$$

Now solve the equation y=\frac{\frac{84}{25}±\frac{4\sqrt{95}i}{5}}{-\frac{128}{25}} when ± is minus. Subtract \frac{4i\sqrt{95}}{5} from \frac{84}{25}.

$$y=\frac{-\frac{4\sqrt{95}i}{5}+\frac{84}{25}}{-\frac{128}{25}}$$

Divide \frac{84}{25}-\frac{4i\sqrt{95}}{5} by -\frac{128}{25} by multiplying \frac{84}{25}-\frac{4i\sqrt{95}}{5} by the reciprocal of -\frac{128}{25}.

$$y=\frac{-21+5\sqrt{95}i}{32}$$

There are two solutions for y: \frac{-21-5i\sqrt{95}}{32} and \frac{-21+5i\sqrt{95}}{32}. Substitute \frac{-21-5i\sqrt{95}}{32} for y in the equation x=-\frac{6}{5}y+\frac{7}{5} to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{6}{5}\times \frac{-5\sqrt{95}i-21}{32}+\frac{7}{5}$$

Multiply -\frac{6}{5} times \frac{-21-5i\sqrt{95}}{32}.

$$x=\frac{-6\times \frac{-5\sqrt{95}i-21}{32}+7}{5}$$

Now substitute \frac{-21+5i\sqrt{95}}{32} for y in the equation x=-\frac{6}{5}y+\frac{7}{5} and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{6}{5}\times \frac{-21+5\sqrt{95}i}{32}+\frac{7}{5}$$

Multiply -\frac{6}{5} times \frac{-21+5i\sqrt{95}}{32}.

$$x=\frac{-6\times \frac{-21+5\sqrt{95}i}{32}+7}{5}$$

The system is now solved.

$$x=\frac{-6\times \frac{-5\sqrt{95}i-21}{32}+7}{5},y=\frac{-5\sqrt{95}i-21}{32}\text{ or }x=\frac{-6\times \frac{-21+5\sqrt{95}i}{32}+7}{5},y=\frac{-21+5\sqrt{95}i}{32}$$