# System of equations

#### ironluis

##### New member
I need your help for solved this.

X^2+y^2=9
(x+3)^2+(y-3)^2=9

#### Jameson

Staff member
Re: I need help!

Hi ironluis,

Welcome to MHB!

Is this a question about systems of equations? Are you supposed to look at the two equations and solve for $x$ and $y$? This can be done algebraically however you posted the question in our geometry forum. Are you supposed to solve it geometrically?

Im sorry

#### Prove It

##### Well-known member
MHB Math Helper
Notice both equations are equal to 9, so they are equal to each other. Set them equal to each other, expand, simplify...

#### MarkFL

Staff member
A more geometric approach would be to consider the point midway between the center of the circles. We use the mid-point in this case because the radii of the circles is the same. If the distance of this midpoint to the radii is greater than the radii, then there is no solution. If this distance is equal to the radii, there is one solution, and if it is less than the radii, and greater than zero, then there are two solutions. If the distance is zero, then the circles are concurrent and there are an infinite number of solutions.

Next, compute the slope of the line segment connecting the center of the circles, and observe that the solutions will lie along the line perpendicular to this segment, and passing through the mid-point of the centers.

This line will give you the result suggested by Prove It's much more straightforward algebraic approach.

Then you want to find the points on this line which satisfy either of the equations describing the circles.

#### johng

##### Well-known member
MHB Math Helper
Hi,
I think the best geometric solution to the problem of intersection of two circles is given at Circle, Cylinder, Sphere
I have used this algorithm with good success.