TITLE: Calculating the value of $\frac{a-d}{b-c}$
QUESTION [5 upvotes]: If $\frac{a-b}{c-d}=2$ and $\frac{a-c}{b-d} = 3$ then determine the value of: 
$$\frac{a-d}{b-c}$$
Where $a,b,c,d$ are real numbers.
Can someone please help me with this and give me a hint? I tried substitutions and solving them simultaneously but I couldn't determine this value. Please help.

REPLY [3 votes]: Cross-multiplying the two equations results in
$$a-b=2(c-d)\text{ (1)}\\a-c=3(b-d)\text{ (2)}$$ 
Eliminating $a$ by subtracting $(1)$ from $(2)$ results in
$$b-c=3b-d-2c\text{ (3)}$$
Eliminating $d$ by subtracting $2\times(1)$ from $3\times(2)$ results in
$$6(b-c)=-2c+3b-a\text{ (4)}$$
Finally subtracting $(3)$ from $(4)$ results in
$$5(b-c)=d-a$$
leading to
$$\frac{a-d}{b-c}=-5$$