Multiply through by :
What did you get when you substituted this in and tried to solve?
solve [(x^2)/25 + (y^2)/4 -25=0, (x^2)/16 + (y^2)/9 -29=0, x, y] - Wolfram|Alpha
I am not certain what the method is. I have made and equation y^2=... from the first equation and put it into the other but I did not get +or-20.
Another way:
x^2/25 + y^2/4 = 25 [1]
x^2/16 + y^2/9 = 29 [2]
[1] *100: 4x^2 + 25y^2 = 2500 [1]
[2] * 144: 9x^2 + 16y^2 = 4176 [1]
36x^2 + 225y^2 = 22500 : [1]*9
-36x^2 - 64y^2 = -16704 : [2]*-9
Add above: 161y^2 = 5796
Now solve for y, then substitute to solve for x