Question

Given that c (ax + by - cz) = b (ax - by + cz) = a ( -ax + by + cz) = 2
then ax + by + cz/ab + bc + ca = ?

Answer 1

Let's unpack the first statement.
c (ax + by - cz) = b (ax - by + cz) = a ( -ax + by + cz) = 2

Answer 2

If we look at the first two statements
c (ax + by - cz) = 2
b (ax - by + cz) = 2

It follows that
(ax + by - cz) = 2/c
(ax - by + cz) = 2/b

now you can add these two equations to eliminate variables x and y

2ax = 2/c + 2/b

Answer 3

can you solve that for x ?

Answer 4

Yes. How do we exactly approach such questions? Is there any trick?

Answer 5

What I wrote is clear so far?

Answer 6

x = 1/ac + 1/ab

Answer 7

Okay

Answer 8

The approach I used is common to solving linear equations, try to find a way to eliminate a variable. Well not exactly this approach, after some work

Answer 9

lets be more careful with solving for x

Answer 10

2ax = 2/c + 2/b
you divided out 2 correctly
ax = 1/c + 1/b
first simplify 1/c + 1/b
ax = ( b + c ) / (bc)
now divide by a
x = ( b + c ) / (abc)

Answer 11

$$ \large \frac{1}{b} + \frac{1}{c} = \frac{1}{b}\cdot \frac {c}{c} + \frac{1}{c} \cdot \frac{b}{b}= \frac{c}{bc} + \frac{b}{bc} = \frac{b+c}{bc}$$

Answer 12

Thank you @perl