Question

Refer to this question below Thank You.

Answer 1

Let's solve this problem using substitution.  Given that x-y=8, x = 8 + y.
(Then x^2 = 64 + 16y + y^2)
This other equation is (x-2)^2 + (y-1)^2 = 25.

Easier to substitute 8 + y for x in (x-2)^2:

(8 + y - 2)^2 + (y-1)^2 = 25

(6 + y)^2 + (y-1)^2 = 25
36 + 12y + y^2 + y^2 - 2y + 1 = 25

Re-writing this in descending powers of y:

2y^2 + 10y + 36 + 1 = 25

Then 2y^2 + 10y - 12 = 0

Reduce by division by 2:  y^2 + 5y - 6 = 0   =   (y+3)(y+2) = 0

Then y=-3 and y=-2.       From each of these we get x:  x = 8 + y

So x = 8 - 3 = 5 and x = 8 - 2 = 6.       There are common solutions.  

Try (5, -3) and (6, -2).  Do these points satisfy both of the given equations?  If they do, you've shown that we have common solutions.