Solving a System of Equations Representing Intersecting Circles

In this article, we will walk through the meticulous process of solving a system of equations that represents two intersecting circles. This method combines algebraic manipulation and geometric interpretation to pinpoint the exact points of intersection. The steps we will follow include rewriting the equations, eliminating variables, substituting, and finally solving a quadratic equation.

Introduction

The given system of equations is:

Equation 1: Equation 2:

Our objective is to determine the points of intersection of these two circles. This process involves a series of algebraic manipulations and the application of the quadratic formula. Let's begin by rewriting and expanding these equations.

Expanding and Rewriting the Equations

Equation 1: [ x^3 - y^2 20 ] Expanding and simplifying, we get:

[ x^3 - y^2 6x - 4y - 7 0 ]

Equation 2: [ x^2 - y^3 2 2 ] Expanding and simplifying, we get:

[ x^2 - y^3 - 4x - 6y 11 0 ]

Eliminating One Variable

To eliminate one variable, we subtract Equation 1 from Equation 2:

[ (x^2 - y^3 - 4x - 6y 11) - (x^3 - y^2 6x - 4y - 7) 0 ]

After simplifying, we obtain:

[ -1 - 2y 18 0 ]

Rearranging this equation, we get:

[ 5x - y 9 ]

Substituting Back

We can now substitute ( y 9 - 5x ) into Equation 1:

[ x^3 - (9 - 5x)^2 6x - 4(9 - 5x) - 7 0 ]

Expanding and simplifying this equation:

[ x^3 - (81 - 9 25x^2) 6x - 36 2 - 7 0 ]

Further simplifying:

[ x^3 - 25x^2 11 - 124 0 ]

Dividing the whole equation by 13, we get:

[ 13x^2 - 38x - 19 0 ]

Using the quadratic formula ( x frac{-b pm sqrt{b^2 - 4ac}}{2a} ), where ( a 13 ), ( b -38 ), and ( c -19 ):

[ x frac{38 pm sqrt{(-38)^2 - 4 cdot 13 cdot (-19)}}{2 cdot 13} ]

Calculating the discriminant:

[ b^2 - 4ac 1444 988 2432 ]

Therefore, we get:

[ x frac{38 pm sqrt{2432}}{26} ]

Further simplifying:

[ x frac{38 pm 2sqrt{608}}{26} ]

[ x frac{38 pm 2sqrt{152}}{26} ]

[ x frac{38 pm 2sqrt{4 cdot 38}}{26} ]

[ x frac{38 pm 4sqrt{38}}{26} ]

[ x frac{19 pm 2sqrt{38}}{13} ]

Finding Corresponding ( y ) Values

Substituting the values of ( x ) back into ( y 9 - 5x ):

For ( x frac{19 2sqrt{38}}{13} ):

[ y 9 - 5 left(frac{19 2sqrt{38}}{13}right) ]

For ( x frac{19 - 2sqrt{38}}{13} ):

[ y 9 - 5 left(frac{19 - 2sqrt{38}}{13}right) ]

The final solutions for ( (x, y) ) will be:

[ (x_1, y_1) ] and [ (x_2, y_2) ] where ( x_1 ) and ( x_2 ) are computed from ( x frac{19 pm 2sqrt{38}}{13} ).

In conclusion, this method precisely determines the points of intersection of the two circles, providing both algebraic and geometric insights into the solution of the system of equations.