Question 113520
It's great to see someone posting their working and, not only that but checking the answer as well, realising they've done something wrong. These are great traits that you should be proud of. Anyway, to the problem, you have only made a very slight mistake:

"If the digits are reversed, the new number increased by 3 equals 4 times the original number"

so 10y+x+3 = <font color="red">4(</font>10x+y<font color="red">)</font>. This tidies up to 6y-39x = -3. Combining this with y=7-x gives 6(7-x)-39x=-3, 42-45x=-3, 45x=45 x=1, y=6.