Answer:

The first thing you should notice is that the order of operations is not "PEMDAS" -- that is, multiplication isn't necessarily done before addition and subtraction.  You can tell this from the first row.  If a product is subtracted from a single-digit number, you can never get 21.

You should pick a row or column with the fewest combinations to start with.  I picked the first column, because it's result is the highest number.  The only two products that are within single-digit range of 53 are 6x8+5 and 5x9+8 (or the first two can be interchanged, giving 8x6+5 and 9x5+8).  It is interesting to note that both the 5 and 8 have to be used in the first column no matter which of these combinations turns out right.  Moreover, the bottom number in the first column is either 5 or 8.  The possibilities are:

The next biggest total is the third row, which totals 31.  The only products in single-digit range of 31 are 8x4, 5x7, 9x4, and 8x5.  These combinations (and their reversals) seem like a whole lot to check, but don't worry -- many can be eliminated right off the bat.  Since both 5 and 8 have to appear in the first column, you can eliminate 8x5.  You can also eliminate 9x4 because the lower-left number has to be either 5 or 8.  This leaves just two possibilities for the bottom row: 8x4-1 and 5x7-4.  Now, the possibilities are:

Turning now to the first row, only 3 x 7 equals 21, so let's look at the four possibilities, above.  In the first possibility, 5-2x7 is the only way the first row can be filled out.

The second possibility allows 9-2x3=21, but then the middle column would be 2+3x4, using the 3 twice, so that's no good.  The second possibility also allows 9-6x3=21, but then the middle column doesn't work, either.  So the second possibility is out.

The third and fourth possibilities can be ruled out because of the middle column -- 7 isn't a factor of 20, so it doesn't matter what the first two numbers of the 2^nd column are; the product can't be 21.

That leaves one possibility for filling out the first and third rows:

Now there are only two numbers that haven't been used -- 3 and 6.  9 divided by 3 times 6 equals 18, so this is the final answer.