To solve the given system of nonlinear equations, we have two equations involving $x$ and $y$:

1. $x^2 + y^2 + 4x - 6y - 13 = 0$
2. $x^2 + y^2 - 8x + 2y - 19 = 0$

Step 1: Subtract the Equations

Let's subtract the second equation from the first to eliminate $x^2$ and $y^2$:

$(x^2 + y^2 + 4x - 6y - 13) - (x^2 + y^2 - 8x + 2y - 19) = 0$

Simplify by combining like terms:

$(4x - (-8x)) + (-6y - 2y) + (-13 + 19) = 0$

$4x + 8x - 6y - 2y + 6 = 0$

$12x - 8y + 6 = 0$

Step 2: Solve for One Variable

Let's solve for $y$ in terms of $x$:

$12x - 8y + 6 = 0$

Rearrange to solve for $y$:

$8y = 12x + 6 \implies y = \frac{12x + 6}{8} = \frac{3x + 3}{2}$

$y = \frac{3}{2}x + \frac{3}{2}$

Step 3: Substitute $y$ into One of the Original Equations

Substitute the expression for $y$ into the first equation:

$x^2 + \left( \frac{3}{2}x + \frac{3}{2} \right)^2 + 4x - 6 \left( \frac{3}{2}x + \frac{3}{2} \right) - 13 = 0$

Expand and simplify:

$x^2 + \left( \frac{9}{4}x^2 + \frac{9}{2}x + \frac{9}{4} \right) + 4x - \left( 9x + 9 \right) - 13 = 0$

Combine like terms:

$x^2 + \frac{9}{4}x^2 + \frac{9}{2}x + \frac{9}{4} + 4x - 9x - 9 - 13 = 0$

$x^2 \left( 1 + \frac{9}{4} \right) + x \left( \frac{9}{2} + 4 - 9 \right) + \left( \frac{9}{4} - 9 - 13 \right) = 0$

Simplify coefficients:

$\frac{13}{4}x^2 - \frac{1}{2}x - \frac{55}{4} = 0$

Multiply everything by 4 to clear fractions:

$13x^2 - 2x - 55 = 0$

Step 4: Solve the Quadratic Equation

Use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 13$, $b = -2$, and $c = -55$:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(13)(-55)}}{2(13)}$

Calculate the discriminant:

$x = \frac{2 \pm \sqrt{4 + 2860}}{26}$

$x = \frac{2 \pm \sqrt{2864}}{26}$

Simplify the square root (since 2864 = 4 \times 716):

$x = \frac{2 \pm 2\sqrt{716}}{26} = \frac{1 \pm \sqrt{716}}{13}$

Step 5: Find the Corresponding $y$ Values

Now, substitute the values of $x$ back into the expression for $y$:

$y = \frac{3}{2}x + \frac{3}{2}$

Substitute $x = \frac{1 + \sqrt{716}}{13}$ and $x = \frac{1 - \sqrt{716}}{13}$ to find the corresponding $y$ values.

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