This problem is just a matter of grouping together like terms.  Remember that terms like $\displaystyle xy$ are treated as though they were their own, different variable:

$\displaystyle 3x + 4x - 3y - 15y + 2xy$

The only part that might be a little hard is:

$\displaystyle -3y - 15y$

If you are confused, think of your number line.  This is like "going back" (more negative) from 15.  Therefore, you ranswer will be:

$\displaystyle 7x + 2xy - 18y$

This problem really is a trick question.  There are no common terms among any of the parts of the expression to be simplified.  In each case, you have an independent variable or set of variables: $\displaystyle x, y, xy,$ and $\displaystyle yz$.  Therefore, do not combine any of the elements!

Remember, when there is a subtraction outside of a group, you should add the opposite of each member.  That is:

$\displaystyle 5x + 3y - (4x + 2y) = 5x + 3y + (-4x) + (-2y)$

That is a bit confusing, so let's simplify.  When you add a negative, you subtract:

$\displaystyle 5x + 3y - 4x - 2y$

Now, group your like variables:

$\displaystyle 5x- 4x + 3y - 2y$

Finally, perform the subtractions and get: $\displaystyle x + y$

Begin by rewriting the subtracted group as a set of added negative numbers:

$\displaystyle 4xy + 3y - (2xy + 3x) - y = 4xy + 3y + (-2xy) + (-3x) - y$

Now, simplify that a little by rewriting the additions of negatives as being mere subtractions:

$\displaystyle 4xy + 3y - 2xy - 3x - y$

Next, move the like terms next to each other:

$\displaystyle 4xy - 2xy + 3y- y - 3x$

Finally, combine like terms:

$\displaystyle 2xy + 2y - 3x$

You need to begin by distributing the minus sign through the whole group $\displaystyle (13b - 2a)$.  This gives you:

$\displaystyle 15a + 23b - (13b - 2a) = 15a + 23b - 13b - (-2a)$

Simplifying the double negative, you get:

$\displaystyle 15a + 23b - 13b + 2a$

Now, you can move the like terms next to each other:

$\displaystyle 15a + 2a + 23b - 13b$

Finally, simplify:

$\displaystyle 17a + 10b$

This problem is as simple as it appears.  All that you need to do is group together like terms:

$\displaystyle 4x - 3x + 3y - 3z + 2$

The only like terms are the $\displaystyle x$ terms.  Therefore, the simple answer is a matter of subtracting 3 from 4:

$\displaystyle x + 3y - 3z + 2$

First, start by distributing the subtraction through the terms in parentheses.  Note that you will be subtracting negative numbers:

$\displaystyle 21x + 51xy - 3x - 2x - (-15xy)$

Subtracting a negative is the same as adding a positive:

$\displaystyle 21x + 51xy - 3x - 2x +15xy$

Now, group the like terms:

$\displaystyle 21x - 3x - 2x+ 51xy +15xy$

All you need to do now is combine like terms:

$\displaystyle 16x + 66xy$

Begin by distributing the subtraction through the parentheses:

$\displaystyle 33x + 25y - 22xy + 2z - 22y - 21x - 4z$

Next, group the like terms:

$\displaystyle 33x - 21x + 25y - 22y - 22xy + 2z - 4z$

Now, combine them:

$\displaystyle 12x + 3y - 22xy -2z$

Begin by putting similar variables together.  Remember that combinations of variables such as $\displaystyle x^2y$ are treated like a separate variable:

$\displaystyle 31x + 21y - 2y + 3x^{2} - 2x^{2} + 2x^{2}y$

Combine like terms:

$\displaystyle 31x + 19y + x^{2} + 2x^{2}y$

You can then rearrange the variables to get the answer as written:

$\displaystyle 31x + x^{2} + 19y + 2x^{2}y$

Begin by distributing the subtraction through the group:

$\displaystyle 53x - 21x - (-33y) + 32y$

Next, change the double negative to a positive:

 $\displaystyle 32x + 65y$