Finding points of Intersections between circles and lines

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Video Guide
0:10 Review of 3 possible intersections of a circle and a line
0:48 Problem 1 Solve by Graphing
2:26 Problem 2 Solve by graphing
3:40 Problem 3
5:10 Problem 4
5:45 Solve the intersection of lines algebraically
6:29 Problem 5 this involves solving the problem algebraically
Follow these steps:
Solve the linear equation
The substitute the linear equation into the equation representing the circle
Solve for a variable
Substitute the variables back into the linear equation to get the second variable
10:00 Problem 6
13:20 Number 8

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[eoinMB]

Hi what if you have a line thats something + 2y -20 = 0? Do you use fractions in this case to solve for x or y?

[thequattro]

How do I use any of this with real coordinates?
There are points for start and end of the line, there is a point for a circle and it's radius.
startX, Y = [10, 10]
endX, Y = [15, 15]
circle_centerX, Y = [12, 12]
circle radius = 2

How do I get intersection coordinates? :D

[luisserranorubio]

how would you find the distance between the two intersections?

[gabadagomes]

you made a mistake folding out -3x+10 squared
=9y (squared)+60y+100