Question 188829
x^2 + y^2 = 169
x^2 - 8y  = 104 
-------------------Subtraction eliminates x^2
y^2 + 8y = 65
A quadratic equation:
y^2 + 8y - 65 = 0
Factors to:
(y + 13)(y - 5) = 0
Two solutions
y = -13
y = +5
;
Find x when y = -13 using the 1st equation:
x^2 + (-13)^2 = 169
x^2 + 169 = 169
x^2 = 169 - 169
x^2 = 0
x = 0
when y = 5
x^2 + 5^2 = 169
x^2 = 169 - 25
x^2 = 144
x = {{{sqrt(144)}}}
x = +/-12
:
:
Check solutions x=0, y=-13, in 2nd equation
0 - 8(-13) = 104
0 + 104 = 104
:
Check solutions x=+/-12, y=5 in 2nd equation
12^2 - 8(5) = 104
144 - 40 = 104
:
Graphing it,we can see the actual solutions are; x=-12, y=5 and x=+12, y=5:
{{{ graph( 300, 200, -20, 20, -20, 20, sqrt(169-x^2), -(104-x^2)/8) }}}