Odpowiedź:
a)
x² + y² = 13
x - y - 4 = 0 ⇒ x = y + 4
(y + 4)² + y² = 13
y² + 8y + 16 + y² = 13
2y² + 8y + 16 - 13 = 0
2y² + 8y + 3 = 0
a = 2 , b = 8 , c = 3
Δ = b² - 4ac = 8² - 4 * 2 * 3 = 64 - 24 = 40
√Δ = √40 = √(4 * 10) = 2√10
y₁ = ( - b - √Δ)/2a = ( - 8 - 2√10)/4 = - 2(4 + √10)/4 = - (4 + √10)/2
y₂ = ( - b + √Δ)/2a = ( - 8 + 2√10)/4 = 2(√10 - 4)/4 = (√10 - 4)/2
x₁ = y₁ + 4 = - (4 + √10)/2 + 4 = (- 4 - √10 + 8)/2 = (4 - √10)/2
x₂ = y₂ + 4 = (√10 - 4)/2 + 4 = (√10 - 4 + 8)/2 = (√10 + 4)/2
b)
x² - y² = 3
x - y = 1 ⇒ x = y + 1
(y + 1)² - y² = 3
y² + 2y + 1 - y² = 3
2y + 1 = 3
2y = 3 - 1 = 2
y = 2/2 = 1
x - y = 1
x - 1 = 1
x = 1 + 1 = 2
c)
xy = 5
x - y = 0 ⇒ x = y
x * x = 5
x² = 5
x² - 5 = 0
(x - √5)(x + √5) = 0
x - √5 = 0 ∨ x + √5 = 0
x₁ = √5 ∨ x₂ = - √5
y₁ = √5 ∨ y₂ = - √5
