Solve {l}{9x^2-25y^2=225}{y+x+4=0}

$$\left\{ \begin{array} { l } { 9 x ^ { 2 } - 25 y ^ { 2 } = 225 } \\ { y + x + 4 = 0 } \end{array} \right.$$

Solve for x, y

$$x = -\frac{25}{4} = -6\frac{1}{4} = -6.25$$

$$y = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

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.

$$y+x+4=0,9x^{2}-25y^{2}=225$$

Solve y+x+4=0 for y by isolating y on the left hand side of the equal sign.

$$y+x+4=0$$

Subtract 4 from both sides of the equation.

$$y+x=-4$$

Subtract x from both sides of the equation.

$$y=-x-4$$

Substitute -x-4 for y in the other equation, 9x^{2}-25y^{2}=225.

$$9x^{2}-25\left(-x-4\right)^{2}=225$$

Square -x-4.

$$9x^{2}-25\left(x^{2}+8x+16\right)=225$$

Multiply -25 times x^{2}+8x+16.

$$9x^{2}-25x^{2}-200x-400=225$$

Add 9x^{2} to -25x^{2}.

$$-16x^{2}-200x-400=225$$

Subtract 225 from both sides of the equation.

$$-16x^{2}-200x-625=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9-25\left(-1\right)^{2} for a, -25\left(-4\right)\left(-1\right)\times 2 for b, and -625 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-200\right)±\sqrt{\left(-200\right)^{2}-4\left(-16\right)\left(-625\right)}}{2\left(-16\right)}$$

Square -25\left(-4\right)\left(-1\right)\times 2.

$$x=\frac{-\left(-200\right)±\sqrt{40000-4\left(-16\right)\left(-625\right)}}{2\left(-16\right)}$$

Multiply -4 times 9-25\left(-1\right)^{2}.

$$x=\frac{-\left(-200\right)±\sqrt{40000+64\left(-625\right)}}{2\left(-16\right)}$$

Multiply 64 times -625.

$$x=\frac{-\left(-200\right)±\sqrt{40000-40000}}{2\left(-16\right)}$$

Add 40000 to -40000.

$$x=\frac{-\left(-200\right)±\sqrt{0}}{2\left(-16\right)}$$

Take the square root of 0.

$$x=-\frac{-200}{2\left(-16\right)}$$

The opposite of -25\left(-4\right)\left(-1\right)\times 2 is 200.

$$x=\frac{200}{2\left(-16\right)}$$

Multiply 2 times 9-25\left(-1\right)^{2}.

$$x=\frac{200}{-32}$$

Reduce the fraction \frac{200}{-32} to lowest terms by extracting and canceling out 8.

$$x=-\frac{25}{4}$$

There are two solutions for x: -\frac{25}{4} and -\frac{25}{4}. Substitute -\frac{25}{4} for x in the equation y=-x-4 to find the corresponding solution for y that satisfies both equations.

$$y=-\left(-\frac{25}{4}\right)-4$$

Multiply -1 times -\frac{25}{4}.

$$y=\frac{25}{4}-4$$

Add -\frac{25}{4}\left(-1\right) to -4.

$$y=\frac{9}{4}$$

The system is now solved.

$$y=\frac{9}{4},x=-\frac{25}{4}\text{ or }y=\frac{9}{4},x=-\frac{25}{4}$$