What's a general method to solve this calculation puzzle?

I need help to work out how to solve this puzzle. The puzzle is very self-explanatory...

• What is the source of this puzzle? Jun 28 at 5:45
• no idea, a fellow mate asked me to solve it Jun 28 at 6:04
• 4 variables and 4 equations, wolframalpha gives a solution $9/10, 21/10,59/10,10$
– z100
Jun 28 at 6:24

This is a system of 4 equations with 4 variables:

$$a+b=8$$
$$a*c=21$$
$$b-d=5$$
$$c*d=9$$

From 2nd and 4th equation:

$$a=21/c$$
$$d=9/c$$

For easier writing let's put $$1/c=x$$ and insert $$a$$ and $$d$$ into 1st and 3rd equation:

$$21x+b=8$$
$$b-9x=5$$
so $$30x = 3$$ or $$x=1/10$$

From here it is easy:

$$c=10$$, $$a=21/10$$, $$d=9/10$$ and $$b=59/10$$