what is a/b?

Question

(a-1)x+(b-1)y=0

(b+1)x+(a+1)y=0

has non trivial solution

find a/b?

Answer 1

(a - 1)x + (b - 1)y = 0

(b + 1)x + (a + 1)y = 0

 

ax - x + by - y = 0

bx + x + ay + y = 0        distribute

 

ax + bx + by + ay = 0     add equations

ax + ay = -bx - by           subtract bx and by from both sides

a(x + y) = -b(x + y)         factor

a = -b                              divide both sides by x + y

a/b = -1                          divide both sides by b

Answer 2

If you sum the two equations you get:

 

(a-1)x+(b+1)x+(b-1)y+(a+1)y=0

 

ax-x+bx+by-y+ay+y=0

 

ax+bx+by+ay=0

 

(a+b)x+(a+b)y=0

 

(a+b)(x+y)=0

 

a+b=0

 

a = -b

 

a/b = -1