Solve {l}{x^2-4y^2=9}{3x+4y=7}

$$\left\{ \begin{array} { l } { x ^ { 2 } - 4 y ^ { 2 } = 9 } \\ { 3 x + 4 y = 7 } \end{array} \right.$$

Solve for x, y

$$x=\frac{17}{5}=3.4\text{, }y=-\frac{4}{5}=-0.8$$

$$x=5\text{, }y=-2$$

Steps Using Substitution and Quadratic Formula

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.

$$3x+4y=7,-4y^{2}+x^{2}=9$$

Solve 3x+4y=7 for x by isolating x on the left hand side of the equal sign.

$$3x+4y=7$$

Subtract 4y from both sides of the equation.

$$3x=-4y+7$$

Divide both sides by 3.

$$x=-\frac{4}{3}y+\frac{7}{3}$$

Substitute -\frac{4}{3}y+\frac{7}{3} for x in the other equation, -4y^{2}+x^{2}=9.

$$-4y^{2}+\left(-\frac{4}{3}y+\frac{7}{3}\right)^{2}=9$$

Square -\frac{4}{3}y+\frac{7}{3}.

$$-4y^{2}+\frac{16}{9}y^{2}-\frac{56}{9}y+\frac{49}{9}=9$$

Add -4y^{2} to \frac{16}{9}y^{2}.

$$-\frac{20}{9}y^{2}-\frac{56}{9}y+\frac{49}{9}=9$$

Subtract 9 from both sides of the equation.

$$-\frac{20}{9}y^{2}-\frac{56}{9}y-\frac{32}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4+1\left(-\frac{4}{3}\right)^{2} for a, 1\times \frac{7}{3}\left(-\frac{4}{3}\right)\times 2 for b, and -\frac{32}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-\frac{56}{9}\right)±\sqrt{\left(-\frac{56}{9}\right)^{2}-4\left(-\frac{20}{9}\right)\left(-\frac{32}{9}\right)}}{2\left(-\frac{20}{9}\right)}$$

Square 1\times \frac{7}{3}\left(-\frac{4}{3}\right)\times 2.

$$y=\frac{-\left(-\frac{56}{9}\right)±\sqrt{\frac{3136}{81}-4\left(-\frac{20}{9}\right)\left(-\frac{32}{9}\right)}}{2\left(-\frac{20}{9}\right)}$$

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

$$y=\frac{-\left(-\frac{56}{9}\right)±\sqrt{\frac{3136}{81}+\frac{80}{9}\left(-\frac{32}{9}\right)}}{2\left(-\frac{20}{9}\right)}$$

Multiply \frac{80}{9} times -\frac{32}{9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{56}{9}\right)±\sqrt{\frac{3136-2560}{81}}}{2\left(-\frac{20}{9}\right)}$$

Add \frac{3136}{81} to -\frac{2560}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\left(-\frac{56}{9}\right)±\sqrt{\frac{64}{9}}}{2\left(-\frac{20}{9}\right)}$$

Take the square root of \frac{64}{9}.

$$y=\frac{-\left(-\frac{56}{9}\right)±\frac{8}{3}}{2\left(-\frac{20}{9}\right)}$$

The opposite of 1\times \frac{7}{3}\left(-\frac{4}{3}\right)\times 2 is \frac{56}{9}.

$$y=\frac{\frac{56}{9}±\frac{8}{3}}{2\left(-\frac{20}{9}\right)}$$

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

$$y=\frac{\frac{56}{9}±\frac{8}{3}}{-\frac{40}{9}}$$

Now solve the equation y=\frac{\frac{56}{9}±\frac{8}{3}}{-\frac{40}{9}} when ± is plus. Add \frac{56}{9} to \frac{8}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{\frac{80}{9}}{-\frac{40}{9}}$$

Divide \frac{80}{9} by -\frac{40}{9} by multiplying \frac{80}{9} by the reciprocal of -\frac{40}{9}.

$$y=-2$$

Now solve the equation y=\frac{\frac{56}{9}±\frac{8}{3}}{-\frac{40}{9}} when ± is minus. Subtract \frac{8}{3} from \frac{56}{9} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{\frac{32}{9}}{-\frac{40}{9}}$$

Divide \frac{32}{9} by -\frac{40}{9} by multiplying \frac{32}{9} by the reciprocal of -\frac{40}{9}.

$$y=-\frac{4}{5}$$

There are two solutions for y: -2 and -\frac{4}{5}. Substitute -2 for y in the equation x=-\frac{4}{3}y+\frac{7}{3} to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{4}{3}\left(-2\right)+\frac{7}{3}$$

Multiply -\frac{4}{3} times -2.

$$x=\frac{8+7}{3}$$

Add -2\left(-\frac{4}{3}\right) to \frac{7}{3}.

$$x=5$$

Now substitute -\frac{4}{5} for y in the equation x=-\frac{4}{3}y+\frac{7}{3} and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{4}{3}\left(-\frac{4}{5}\right)+\frac{7}{3}$$

Multiply -\frac{4}{3} times -\frac{4}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{16}{15}+\frac{7}{3}$$

Add -\frac{4}{3}\left(-\frac{4}{5}\right) to \frac{7}{3}.

$$x=\frac{17}{5}$$

The system is now solved.

$$x=5,y=-2\text{ or }x=\frac{17}{5},y=-\frac{4}{5}$$