Answer:
(3, 2 ) and (17, - 12 )
Step-by-step explanation:
Given the 2 equations
x + y = 5 → (1)
x² - 2y² = 1 → (2)
Rearrange (1) expressing y in terms of x by subtracting x from both sides
y = 5 - x → (3)
Substitute y = 5 - x into (2)
x² - 2(5 - x)² = 1 ← expand and simplify left side
x² - 2(25 - 10x + x²) = 1
x² - 50 + 20x - 2x² = 1
- x² + 20x - 50 = 1 ( subtract 1 from both sides )
- x² + 20x - 51 = 0 ← multiply through by - 1
x² - 20x + 51 = 0 ← in standard form
(x - 17)(x - 3) = 0 ← in factored form
Equate each factor to zero and solve for x
x - 17 = 0 ⇒ x = 17
x - 3 = 0 ⇒ x = 3
Substitute these values into (3) for corresponding values of y
x = 17 : y = 5 - 17 = - 12 ⇒ (17, - 12 )
x = 3 : y = 5 - 3 = 2 ⇒ (3, 2 )
Solutions are (3, 2 ) and (17, - 12 )
