The different columns aren't really equations here. (THere's no equal sign, so they can't be.)
Instead we have the integer i divided by 3 with a remainder of 1. Thus, we start with an integer, multiply it by 3 and add 1 to it to arrive at i.
the integer (they're calling it k) multiplied by 3 is 3k. When we add1 to it, we get 3k+1. Those are the first two columns.
They they say that when i is divided by 4, we get a remainder of 3. Same process here. Take an integer, multiply it by 4, and then add 3 to it. Those are the third and fourth column.
THey start with value of k as 2. Multiply it by 3 and we get 6. Add one more and we get 7. That's a possible value for i. (We don't start with k=1 here because we'd be outside the acceptable range of "greater than 5 but less than 30")
Then start with value of m=1, multiply it by 4 and we get 4, and add 3 and we get 7. That's a possible value for i as well.
We need to find a situation where we have a value that works in both situations. 19 shows up both in the 3k+1 column and the 4m+3 column, so it's our answer.