Solve {l}{9x^2-4y^2=36}{x-y=2}

$$\left\{ \begin{array} { l } { 9 x ^ { 2 } - 4 y ^ { 2 } = 36 } \\ { x - y = 2 } \end{array} \right.$$

Solve for x, y

$$x=2\text{, }y=0$$

$$x=-\frac{26}{5}=-5.2\text{, }y=-\frac{36}{5}=-7.2$$

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.

$$x-y=2,-4y^{2}+9x^{2}=36$$

Solve x-y=2 for x by isolating x on the left hand side of the equal sign.

$$x-y=2$$

Subtract -y from both sides of the equation.

$$x=y+2$$

Substitute y+2 for x in the other equation, -4y^{2}+9x^{2}=36.

$$-4y^{2}+9\left(y+2\right)^{2}=36$$

Square y+2.

$$-4y^{2}+9\left(y^{2}+4y+4\right)=36$$

Multiply 9 times y^{2}+4y+4.

$$-4y^{2}+9y^{2}+36y+36=36$$

Add -4y^{2} to 9y^{2}.

$$5y^{2}+36y+36=36$$

Subtract 36 from both sides of the equation.

$$5y^{2}+36y=0$$

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

$$y=\frac{-36±\sqrt{36^{2}}}{2\times 5}$$

Take the square root of 36^{2}.

$$y=\frac{-36±36}{2\times 5}$$

Multiply 2 times -4+9\times 1^{2}.

$$y=\frac{-36±36}{10}$$

Now solve the equation y=\frac{-36±36}{10} when ± is plus. Add -36 to 36.

$$y=\frac{0}{10}$$

Divide 0 by 10.

$$y=0$$

Now solve the equation y=\frac{-36±36}{10} when ± is minus. Subtract 36 from -36.

$$y=-\frac{72}{10}$$

Reduce the fraction \frac{-72}{10} to lowest terms by extracting and canceling out 2.

$$y=-\frac{36}{5}$$

There are two solutions for y: 0 and -\frac{36}{5}. Substitute 0 for y in the equation x=y+2 to find the corresponding solution for x that satisfies both equations.

$$x=2$$

Now substitute -\frac{36}{5} for y in the equation x=y+2 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\frac{36}{5}+2$$

Add -\frac{36}{5} to 2.

$$x=-\frac{26}{5}$$

The system is now solved.

$$x=2,y=0\text{ or }x=-\frac{26}{5},y=-\frac{36}{5}$$