Find the point of intersection, if any , between each circle and line with the equations given?

(x+4)^2 + (y-3)^2 = 25: y = x + 2        sub the second equation into the first for y

(x + 4)^2  + ( x + 2 - 3)^2  = 25

(x + 4)^2 + (x - 1)^2  = 25     expand the left side

x^2 + 8x + 16 + x^2 - 2x + 1   = 25     simplify

2x^2 + 6x + 17 = 25      subtract 25 from both sides

2x^2 + 6x - 8  = 0      divide through by 2

x^2 + 3x - 4  =  0     factor

(x - 1) ( x + 4)  = 0       and setiing each factor to 0, we have that  x  = 1  and x = -4

So when x = 1, y= 1 + 2 = 3     and when x = -4,  y= -4 + 2  = -2

So....the intersection points are  (1, 3)  and  ( -4, -2)

x^2 + y^2 = 25: y = x     

Put the second equation into the first for y   and we have

x^2 + x^2  = 25

2x^2  = 25    divide both sides by 2

x^2  = 25/2          take the square root of both sides

x^2  = [ +/-] 5/sqrt( 2)

And since x = y   the intersection points are :

(5/sqrt(2), 5/sqrt(2) )  and   (-5/sqrt(2), -5/sqrt(2) )