Find the interserction points of: The circle, x^2+(y-1)^2=18 and the line, y=x+1.

To solve this question, we have to use substitution. We substitute the line equation,y=x+1, into the circle equation so that we get, x^2 + (x+1-1)^2=18. This reduces to 2x^2 = 18, then dividing both sides by 2 gives us, x^2=9. We then have two values for x, x1=3 and x2=-3. Now to solve for the corresponding y co-ordinates, we can substitute our x co-ordinates into either one of the equations and then check the solution in the other. Substituting x1=3 into the line equation gives us y1=3+1=4, and checking in the circle equation gives us (3)^2 + (4-1)^2 =9+9=18, so it works for x1=3,y1=4. Substituting x2=-3 into the line equation gives us y2=-3+1=-2, and checking in the circle equation gives us (-3)^2 + (-2-1)^2 =9+9=18, so it works for x2=-3, y2=-2.We get that our co-ordinates are (x1,y1)=(3,4) and (x2,y2)=(-3,-2).