Solve {l}{2x^2-y^2=9}{x+y=18}

$$\left. \begin{array} { l } { 2 x ^ { 2 } - y ^ { 2 } = 9 } \\ { x + y = 18 } \end{array} \right.$$

Solve for x, y

$$x=3\sqrt{73}-18\approx 7.632011236\text{, }y=36-3\sqrt{73}\approx 10.367988764$$

$$x=-3\sqrt{73}-18\approx -43.632011236\text{, }y=3\sqrt{73}+36\approx 61.632011236$$

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.

$$x+y=18,-y^{2}+2x^{2}=9$$

Solve x+y=18 for x by isolating x on the left hand side of the equal sign.

$$x+y=18$$

Subtract y from both sides of the equation.

$$x=-y+18$$

Substitute -y+18 for x in the other equation, -y^{2}+2x^{2}=9.

$$-y^{2}+2\left(-y+18\right)^{2}=9$$

Square -y+18.

$$-y^{2}+2\left(y^{2}-36y+324\right)=9$$

Multiply 2 times y^{2}-36y+324.

$$-y^{2}+2y^{2}-72y+648=9$$

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

$$y^{2}-72y+648=9$$

Subtract 9 from both sides of the equation.

$$y^{2}-72y+639=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1+2\left(-1\right)^{2} for a, 2\times 18\left(-1\right)\times 2 for b, and 639 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 639}}{2}$$

Square 2\times 18\left(-1\right)\times 2.

$$y=\frac{-\left(-72\right)±\sqrt{5184-4\times 639}}{2}$$

Multiply -4 times 639.

$$y=\frac{-\left(-72\right)±\sqrt{5184-2556}}{2}$$

Add 5184 to -2556.

$$y=\frac{-\left(-72\right)±\sqrt{2628}}{2}$$

Take the square root of 2628.

$$y=\frac{-\left(-72\right)±6\sqrt{73}}{2}$$

The opposite of 2\times 18\left(-1\right)\times 2 is 72.

$$y=\frac{72±6\sqrt{73}}{2}$$

Now solve the equation y=\frac{72±6\sqrt{73}}{2} when ± is plus. Add 72 to 6\sqrt{73}.

$$y=\frac{6\sqrt{73}+72}{2}$$

Divide 72+6\sqrt{73} by 2.

$$y=3\sqrt{73}+36$$

Now solve the equation y=\frac{72±6\sqrt{73}}{2} when ± is minus. Subtract 6\sqrt{73} from 72.

$$y=\frac{72-6\sqrt{73}}{2}$$

Divide 72-6\sqrt{73} by 2.

$$y=36-3\sqrt{73}$$

There are two solutions for y: 36+3\sqrt{73} and 36-3\sqrt{73}. Substitute 36+3\sqrt{73} for y in the equation x=-y+18 to find the corresponding solution for x that satisfies both equations.

$$x=-\left(3\sqrt{73}+36\right)+18$$

Now substitute 36-3\sqrt{73} for y in the equation x=-y+18 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\left(36-3\sqrt{73}\right)+18$$

The system is now solved.

$$x=-\left(3\sqrt{73}+36\right)+18,y=3\sqrt{73}+36\text{ or }x=-\left(36-3\sqrt{73}\right)+18,y=36-3\sqrt{73}$$