Question

What are the solutions of the following system?

StartLayout Enlarged left-brace 1st row x squared + y squared = 25 2nd row 2x + y = 25 EndLayout
(0, –5) and (–5, 5)
(0, –5) and (5, –15)
(0, –5) and (–4, 3)
(0, –5) and (4, –13)

Answer 1

If the system is exactly
x^2 + y^2 = 25
2x + y = 25
then substitute y = 25 - 2x into the circle:
x^2 + (25 - 2x)^2 = 25
→ 5x^2 - 100x + 600 = 0
→ x^2 - 20x + 120 = 0,
whose discriminant is 400 − 480 = −80 < 0, so there are no real solutions (none of the given pairs satisfies both equations).

If the intended linear equation was 2x + y = −5 (a common small typo), then y = −5 − 2x and
x^2 + (−5 − 2x)^2 = 25
→ 5x(x + 4) = 0
so x = 0 or x = −4, giving the solutions (0, −5) and (−4, 3).