Solve the system using the method of elimination?

Question

x^2 + y^2 = 100

3x-y = 10

How do you do it using the method of elimination?

Answer 1

Given equations are

x² + y² = 100

3x - y = 10

Apply square both sides.

(3 x - y)² = 10²

9x² + y² - 6xy = 100

8x² + x² + y² - 6xy - 100 = 0

Substitute : x² + y² = 100

8x² +100 - 6xy - 100 = 0

8x² - 6xy = 0

Substitute : 3 x - y = 10 ⇒ y = 3x - 10

8x² - 6x(3x - 10) = 0

8x² - 18x² + 60x = 0

- 10x² + 60x = 0

60x - 10x² = 0

10x( 6 - x) = 0

By using zero product property : If AB = 0 then A = 0 , B = 0.

10x  = 0  and  ( 6 - x) = 0

x = 0  and   x = 6

If x = 0

3 x - y = 10 ⇒ y = - 10

If x = 6

3 x - y = 10 ⇒ y = 10 - 3*6 = - 8

Solution :

Given equations are two solutions

1) x = 0 , y = - 10

2) x = 6 , y = - 8