Verify if the lines are intercepting? y = (5-x)/(1+x) y*x^2+x*y^2=6

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll multiply the 1st equation by 1+x both sides:

y*(1+x) = 5-x

We'll remove the brackets:

y + x*y = 5-x

We'll move all terms to one side:

y+x + x*y - 5 = 0 (3)

We'll factorize th second equation by x*y:

x*y(x+y) = 6 (4)

We'll substitute x+y = S and x*y = P

We'll re-write the equations (3) and (4):

S+P-5=0 => S = 5-P

S*P=6

(5-P)*P=6

We'll remove the brackets:

-P^2 + 5P - 6 = 0

P^2  - 5P + 6 = 0

P1 = 1 => S1 = 5-1=4

P2 = 5 => S2 = 5-5 = 0

x + y = S1 <=> x+y = 4

x*y= P1 <=> x*y = 1

x^2 - Sx + P = 0

x^2 - 4x + 1 = 0

x = [4+sqrt(16-4)]/2

x = [4+sqrt(12)]/2

x = (4+2sqrt3)/2

x = 2+sqrt3 ; y = 2-sqrt3

x = 2-sqrt3 ; y = 2+sqrt3

x + y = S2 <=> x+y = 0

x*y= P2 <=> x*y = 5

x^2  + 5 = 0 impossible!

There are no real values of x to satisfy the equation.

The lines are intercepting in the points (2+sqrt3 ; 2-sqrt3) and (2-sqrt3 ; 2+sqrt3).