Question

If 3x+4y=11 and xy=2 solve simultaneously​

Answer 1

(1, 2 ) and (
`(8)/(3)`,
`(3)/(4)` )

Explanation:

3x + 4y = 11 → (1)

xy = 2 ( divide both sides by y , y ≠ 0 )

x =
`(2)/(y)` → (2)

Substitute x =
`(2)/(y)` into (1)

3(
`(2)/(y)` ) + 4y = 11

`(6)/(y)` + 4y = 11 ( multiply through by y )

6 + 4y² = 11y ( subtract 11y from both sides )

4y² - 11y + 6 = 0

Consider the factors of the product of the y² term and the constant term which sum to give the coefficient of the y- term

product = 4 × 6 = 24 and sum = - 11

The factors are - 8 and - 3

Use these factors to split the y- term

4y² - 8y - 3y + 6 = 0 ( factor first/second and third/fourth terms )

4y(y - 2) - 3(y - 2) = 0 ← factor out (y - 2) from each term

(y - 2)(4y - 3) = 0

Equate each factor to zero and solve for y

4y - 3 = 0 ⇒ 4y = 3 ⇒ y =
`(3)/(4)`

y - 2 = 0 ⇒ y = 2

Substitute these values into (2)

y = 2 : x =
`(2)/(2)` = 1 ⇒ (1, 2 )

y =
`(3)/(4)` : x =
`(2)/((3)/(4) )` = 2 ×
`(4)/(3)` =
`(8)/(3)` ⇒ (
`(8)/(3)`,
`(3)/(4)` )