Show that (2 / 3, 2 / 3) is the only solution of {3x + 3y = 4, 9xy = 4}. As well, show that the line and the curve hit each other at (2 / 3, 2 / 3).

3x + 3y = 4

9xy = 4

3y = 4 - 3x

y = (4 - 3x) / 3

9x(4 - 3x) / 3 = 4

9x² - 12x + 4 = 0

(3x - 2)² = 0

x = 2 / 3

y = {4 - 3(2 / 3)} / 3

y = 2 / 3

only solution as (2 / 3, 2 / 3)

tangent to 9xy = 4 at (2 / 3, 2 / 3) as 3x + 3y = 4

 

Show that (2 / 5, 2 / 5) is the only solution of {5x + 5y = 4, 25xy = 4}. As well, show that the line and the curve hit each other at (2 / 5, 2 / 5).

5x + 5y = 4

25xy = 4

5y = 4 - 5x

y = (4 - 5x) / 5

25x(4 - 5x) / 5 = 4

25x² - 20x + 4 = 0

(5x - 2)² = 0

x = 2 / 5

y = {4 - 5(2 / 5)} / 5

y = 2 / 5

only solution as (2 / 5, 2 / 5)

tangent to 25xy = 4 at (2 / 5, 2 / 5) as 5x + 5y = 4

 

Show that (3 / 5, 3 / 5) is the only solution of {5x + 5y = 6, 25xy = 9}. As well, show that the line and the curve hit each other at (3 / 5, 3 / 5).

5x + 5y = 6

25xy = 9

5y = 6 - 5x

y = (6 - 5x) / 5

25x(6 - 5x) / 5 = 9

25x² - 30x + 9 = 0

(5x - 3)² = 0

x = 3 / 5

y = {6 - 5(3 / 5)} / 5

y = 3 / 5

only solution as (3 / 5, 3 / 5)

tangent to 25xy = 9 at (3 / 5, 3 / 5) as 5x + 5y = 6