Question

If ab + xy - xb = 0 and bc + yz - cy = 0, then
what is x/a+c/z equal to ?
​

Answer 1

ANSWER

Given system of eqautions has non-trivial solution.

⇒

∣

∣

∣

∣

∣

∣

∣

∣

a

b

−1

a

−1

c

−1

b

c

∣

∣

∣

∣

∣

∣

∣

∣

=0

⇒a(−c−bc)−a(bc+b)−(bc−1)=0

⇒ab+bc+ca+2abc=1 -----(1)

Now,

1+a

1

+

1+b

1

+

1+c

1

=

(1+a)(1+b)(1+c)

(1+b)(1+c)+(1+a)(1+c)+(1+a)(1+b)

=

1+a+b+c+ab+bc+ca+abc

3+2a+2b+2c+ab+bc+ca

rewriting numerator using (1) gives

=

1+a+b+c+ab+bc+ca+abc

2(1+a+b+c+ab+bc+ca+abc)

=2