Certainly! Let's walk through solving the system of equations step-by-step.
The given system of equations is:
1. [tex]\( x^2 + y^2 = 25 \)[/tex]
2. [tex]\( x^2 + y^2 - 10x - 8y = -32 \)[/tex]
First, let's observe the equations. The first equation represents a circle centered at the origin [tex]\((0,0)\)[/tex] with a radius of 5. The second equation can be rearranged to better understand its geometric meaning.
We start by isolating common terms in the second equation:
[tex]\[ x^2 + y^2 - 10x - 8y = -32 \][/tex]
We notice that [tex]\( x^2 + y^2 \)[/tex] is common in both equations. Let’s manipulate the second equation to find its form and see if it represents another circle. We'll complete the square for the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms in the second equation.
Rewrite the second equation grouping [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms:
[tex]\[ x^2 - 10x + y^2 - 8y = -32 \][/tex]
Complete the square for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 10x = (x - 5)^2 - 25 \][/tex]
Complete the square for [tex]\( y \)[/tex]:
[tex]\[ y^2 - 8y = (y - 4)^2 - 16 \][/tex]
Substitute these completed squares back into the second equation:
[tex]\[ (x - 5)^2 - 25 + (y - 4)^2 - 16 = -32 \][/tex]
[tex]\[ (x - 5)^2 + (y - 4)^2 - 41 = -32 \][/tex]
Move [tex]\( -41 \)[/tex] to the right:
[tex]\[ (x - 5)^2 + (y - 4)^2 = 9 \][/tex]
Now we see that the second equation represents another circle, centered at [tex]\((5, 4)\)[/tex] with a radius of 3.
Now we proceed to solve the system of two circles:
1. [tex]\( x^2 + y^2 = 25 \)[/tex]
2. [tex]\( (x - 5)^2 + (y - 4)^2 = 9 \)[/tex]
To find the intersections of these two circles, we solve these equations simultaneously.
Expanding the second equation:
[tex]\[ (x - 5)^2 + (y - 4)^2 = 9 \][/tex]
[tex]\[ x^2 - 10x + 25 + y^2 - 8y + 16 = 9 \][/tex]
Simplifying, we get:
[tex]\[ x^2 - 10x + y^2 - 8y + 41 = 9 \][/tex]
[tex]\[ x^2 + y^2 - 10x - 8y = -32 \][/tex]
Here, the equations simplify, as previously shown.
The intersections of these two circles are given by:
[tex]\[ \left(\frac{285}{82} - \frac{2\sqrt{851}}{41}, \frac{5\sqrt{851}}{82} + \frac{114}{41}\right) \][/tex]
[tex]\[ \left(\frac{285}{82} + \frac{2\sqrt{851}}{41}, \frac{114}{41} - \frac{5\sqrt{851}}{82}\right) \][/tex]
Therefore, the solutions to the system of equations are:
1. [tex]\( \left( \frac{285}{82} - \frac{2\sqrt{851}}{41}, \frac{5\sqrt{851}}{82} + \frac{114}{41} \right) \)[/tex]
2. [tex]\( \left( \frac{285}{82} + \frac{2\sqrt{851}}{41}, \frac{114}{41} - \frac{5\sqrt{851}}{82} \right) \)[/tex]
These coordinates are the points of intersection of the two circles, hence the solutions to the system of equations.
