Simultaneously
x2 - y2 = 24
x = 3 + 2y

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)

step-by-step explanation:

f