Example 1: Find the points of intersection of the circle with the line given vy their equations
(x  2)^{2} + (y + 3)^{2} = 4
2x + 2y = 1
Solution to Example 1:

We first solve the linear equation for y as follows:
y =  x  1/2

We now substitute y in the equation of the circle by  x  1/2 as follows
(x  2)^{2} + ( x  1/2 + 3)^{2} = 4

We now expand the above equation and group like terms
2 x^{2}  9 x + 25/4 = 0

Solve the above quadratic equation for x to obtain two solutions
x = (9 + √(31)) / 4 and x = (9  √(31)) / 4

We now substitute the values of x already obtained into the equation y =  x  1/2 to obtain the values for y as follows
y = (11  √31) / 4
and y = (11 + √31) / 4

The two points of intersection of the two cirlces are given by
((9 + √(31)) / 4 , (11  √31) / 4 ) and ((9  √(31)) / 4 , (11 + √31) / 4)
Approximated as:(3.64 ,  4.14 ) and (0.86 , 1.36)
Shown below is the graph of the circle, the line and the two points of intersection.
More links and references related to the above topics.