Question

The line y - 3 = 2x intersects the curve 20 + xy = 5y at P and Q. Calculate the coordinates of P and Q.​

Answer 1

see explanation

Explanation:

Given the 2 equations

y - 3 = 2x → (1)

20 + xy = 5y → (2)

Rearrange (1) expressing y in terms of x by adding 3 to both sides

y = 2x + 3 → (3)

Substitute y = 2x + 3 into (2)

20 + x(2x + 3) = 5(2x + 3) ← distribute parenthesis

20 + 2x² + 3x = 10x + 15 ← subtract 10x + 15 from both sides

2x² - 7x + 5 = 0 ← in standard form

(x - 1)(2x - 5) = 0 ← in factored form

Equate each factor to zero and solve for x

x - 1 = 0 ⇒ x = 1

2x - 5 = 0 ⇒ 2x = 5 ⇒ x =
`(5)/(2)`

Substitute each of these values into (3) for corresponding values of y

x = 1 : y = 2 + 3 = 5 ⇒ P(1, 5 )

x =
`(5)/(2)` : y = 5 + 3 = 8 ⇒ Q(
`(5)/(2)`, 8 )