What approach to take to solve this arithmetic or algebraic puzzle?

Question

I came across this puzzle.

I tried to solve it using pen and paper. But I was not even sure whether this puzzle is genuine and has a proper solution. I wrote a program to find the solution. The program found few solutions.

Without the aid of a computer program, how would you go about solving such problems?

I started by labeling the empty columns: a, b, c, d, e, f, g and h.

Then I wrote down some rules:

row_1 = (a + b ) * c = 20

row_2 = (d * 8 * e) = 48

row_3 = (f * g) - h = 38

col_1 = a + d - f = 0

col_2 = b * 8 * g = 80

col_3 = c - e + h = 10

Then I limited the possible values for the variables. For example, d and e can be one of 1, 2, 3, 6. Since 48 / 8 = 6. 1, 2, 3, 6 are divisors of 6.

But I do not know a method to take it further from here to find the solution.

Answer 1

First row: $10$, $2$, $5$

Second row: $-2$, $8$, $-3$

Third row: $8$, $5$, $2$

EDIT:

Daniel Mathias has provided a link that says we must use the numbers $1$ through $9$ and we evaluate from left to right and from top to bottom, ignoring precedence rules.

This gives us

$3\ 2\ 4$

$6\ 8\ 1$

$9\ 5\ 7$