Answer:
The solutions of the system are (5, 1) and (-7, -5)
Step-by-step explanation:
The factorization of a² - b² = (a + b)(a - b)
Let us use this rule to solve the question
∵ x² - y² = 24
→ Factorize the left side
∴ (x + y)(x - y) = 24 ⇒ (1)
∵ x = 3 + 2y → (2)
→ Substitute x in equation (1) by equation (2)
∵ (3 + 2y + y)(3 + 2y - y) = 24
∴ (3 + 3y)(3 + y) = 24
→ Take 3 from the first bracket as a common factor
∵ 3 + 3y = 3(1 + y)
∴ 3(1 + y)(3 + y) = 24
→ Divide both sides by 3
∴ (1 + y)(3 + y) = 8 ⇒ (3)
→ Multiply the two brackets on the left side
∵ (1 + y)(3 + y) = (1)(3) + (1)(y) + (y)(3) + (y)(y)
∴ (1 + y)(3 + y) = 3 + y + 3y + y²
∴ (1 + y)(3 + y) = 3 + 4y + y²
→ Substitute it in the equation (3)
∴ 3 + 4y + y² = 8
→ Subtract 8 from both sides
∵ 3 - 8 + 4y + y² = 8 - 8
∴ -5 + 4y + y² = 0
→ Arrange the left side from greatest power of y
∵ y² + 4y - 5 = 0
→ Factorize it into two factors
∴ (y - 1)(y + 5) = 0
→ Equate each bracket by 0 to find y
∵ y - 1 = 0 ⇒ Add 1 to both sides
∴ y - 1 + 1 = 0 + 1
∴ y = 1
∵ y + 5 = 0 → Subtract 5 from both sides
∴ y + 5 - 5 = 0 - 5
∴ y = -5
→ Substitute the values of y in equation (2) to find x
∵ x = 3 + 2(1) = 3 + 2
∴ x = 5
∵ x = 3 + 2(-5) = 3 + -10
∴ x = -7
∴ The solutions of the system are (5, 1) and (-7, -5)
