I have a problem asking to find two numbers that have a difference of $4$ and product of $22$. So I set up two equations, $s - t = 4$ and $s\times t = 22$. I then proceeded to solve the first equation for $t$ and got $-t = 4 - s$, I then multiplied both sides by $-1$ and got $t = -4 + s$. The solution in the book I am using solved for $s$ and got $s = t + 4$. Are these two equations equivalent or am I doing something wrong when multiplying both sides by $-1$? The answer to the problem was a negative and a positive number but I got two positives instead.
You have done well, multiplication by $-1$ is fine. Now, as you want to compare with your book, write down the expression for $s$, in terms of $t$ and 4 i.e., keep the $s$ at the right hand side and move the $-4$ to the left hand side to get $t+4=s$.
Now, as for the solution, surely both the number should be of same sign, otherwise, if they have opposite sign, their product would be negative. It looks like both the numbers can be positive (as well as negative).
$4$ and $8$ have a difference of 4 but $s-t=-4$ and that's not $4$. So I think your first equation should be $|s-t|=4$.