Find All Possible Natural Roots of a System of Equations
Algebra math problem: How to solve system of two equations?
If x + y = x² and x − y = 1/x² then find all possible natural roots of x and y
x+y=x^2
x-y=\dfrac{1}{x^2}
(x, y)=??
Solution
Let
x + y = x2………………………eq(1)
x − y = 1/x2……………………eq(2)
Add equation 1 and equation 2, then
x + y + x − y = x2 + 1/x2
⇒ 2x = x2 + 1/x2
⇒ 2x3 = x4 + 1
⇒ x4 − 2x3 + 1 = 0
⇒ x4 − x3 − x3 + 1 = 0
⇒ x4 − x3 − x3 + 1 = 0
⇒ x3 (x − 1) − (x3 – 1) = 0
⇒ x3(x − 1) − (x − 1) (x2 + x + 12) = 0
⇒ (x − 1) (x3 − x2 − x − 1) = 0
Here, x − 1 = 0 or x3 − x2 − x − 1 = 0
When x − 1 = 0, then
x = 1 → Natural Solution
x3 − x2 − x − 1 → Has no natural Solution
So x = 1 is the only natural solution
From equation 1
When x = 1
x + y = x2
⇒ 1 + y = 1
⇒ y = 0
Then, natural roots of (x, y) is (1, 0)