Question

Find the number of ordered pairs of positive integers (x,y) that satisfy x2−xy+y2=49

Answer 1

Answer:

Solution set = { (7 , 0) , (0 , 7) , (3 , 8) , (5 , 8) , (8 , 3) , (8 , 5) }

Step-by-step explanation:

We are given that

x² - xy + y² = 49

Putting the y = 3 we get

x² - x(3) + 3² = 49

⇒ x² - 3x + 9 = 49

⇒ x² - 3x + 9 - 49 = 0

⇒ x² - 3x - 40 = 0

⇒ x² - 8x + 5x - 40 = 0

⇒ x(x-8) + 5(x - 8) = 0

⇒ (x - 8)(x+ 5) = 0

⇒ x = 8   or x = -5

We are interested only in positive solution solutions so

(x ,y) = (8 , 3)      is the solution

Similarly

when y = 5 then x =8

So

(x , y) = (8 , 5)      is also the solution

Similarly

When y = 8 then x = 3

So

(x , y) = (3 , 8)      is another solution

Similarly

When y = 8 then x = 5

So

(x , y) = (5 , 8)      is another solution

Similarly

When x = 0  then y = 7

So

(x , y) = (0 , 7)      is another solution

Similarly

When y = 0 then x = 7

So

(x , y) = (7 , 0)      is another solution

Thus

All ordered pairs which are positive are following

Solution set = { (7 , 0) , (0 , 7) , (3 , 8) , (5 , 8) , (8 , 3) , (8 , 5) }

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