Solving a System of Equations: Circle and Line
This article will guide you through solving a system of equations involving a circle and a line. We will analyze the equations (x3)^2 + (y+2)^2 = 16 and 2x + 2y = 10 to find their points of intersection.
Understanding the Equations

(x3)^2 + (y+2)^2 = 16 represents a circle with:
 Center: (3, 2)
 Radius: 4 (square root of 16)

2x + 2y = 10 represents a line. To understand it better, we can rewrite it in slopeintercept form (y = mx + c):
 Slope (m): 1
 Yintercept (c): 5
Solving the System
To find the points of intersection, we need to find the values of x and y that satisfy both equations simultaneously. We can achieve this through the following steps:

Substitution Method:
 Solve the linear equation (2x + 2y = 10) for one variable (let's solve for x):
 x = 5  y
 Substitute this expression for x into the circle equation:
 (5  y  3)^2 + (y + 2)^2 = 16
 Simplify and solve the resulting quadratic equation.
 Solve the linear equation (2x + 2y = 10) for one variable (let's solve for x):

Solving the Quadratic Equation:
 Expanding the equation, we get:
 (2  y)^2 + (y + 2)^2 = 16
 4  4y + y^2 + y^2 + 4y + 4 = 16
 2y^2 = 8
 y^2 = 4
 y = ±2
 Expanding the equation, we get:

Finding the Corresponding x Values:
 Substitute the values of y back into the equation x = 5  y:
 For y = 2, x = 5  2 = 3
 For y = 2, x = 5  (2) = 7
 Substitute the values of y back into the equation x = 5  y:
Conclusion
Therefore, the points of intersection between the circle (x3)^2 + (y+2)^2 = 16 and the line 2x + 2y = 10 are:
 (3, 2)
 (7, 2)
This means the line intersects the circle at two distinct points.