Collecting Like Terms

1. Collecting Like Terms orms ?

2. A term in math is a piece of an expression separated by a plus or minus sign. In the expression 3x + 4y – 7 There are three terms.

3. We know what a term is now, so what are like terms? Like terms are terms that have identical variable pieces. In other words, the letters match.

4. x, y, and z are all different variables. So each of these is a different term. x and x are the same, so they are called like terms. z and z would also be like terms.

5. Let’s forget about math for a few minutes. Instead of talking about terms let’s talk about worms. “Why worms?” You ask. Because they’re so cute.

6. See.

7. We have a box of worms.Some are brown and some are green.

8. We have two types of worms. Let’s collect the worms into groups by color.

9. How many of each color do we have? 3 2

10. Back to math. Instead of worms, now we have terms. Instead of grouping by colors we group by the same variables. (like terms) x y x y x

11. Just imagine the letters are worms and collect them like before. x y x x y

12. Now count up how many of each we have. x y x 3 x y 2

13. Let’s look at this as an expression. Just as before, count up how many of each term you have. x + x + y + x + y 3 x’s and 2 y’s We write this as 3x + 2y

14. But worms are social creatures. Well, not really, but what if they’re traveling in groups instead of individually?

15. In this case we take the number in each group and combine them. 4 3 3

16. Now combine like worms, brown with brown, and yellow with yellow. Since there is only one group of yellow it does not combine with any other worms. 4 7 3 3 The expression is now: 7 3

17. Let’s look at some examples. • 3x + 4y + 2x • 7x + 2y + 6x + 3y • 12z + 14z + 13z = 5x + 4y = 13x + 5y = 39z

18. Collecting like terms(worms) can also be used when reducing a group. We can subtract or add negative amounts. For example, on the expression 4x + 3y + 2x we add the like terms to get 6x + 3y. However, on the expression 4x + 3y – 2x we subtract the x’s to get 2x +3y.

19. The trick to subtraction is to remember that each number goes with the sign or operation that comes before it. 4x + 7y – 3x – 2y + 3z – 2y Now collect like terms: x + 3y + 3z

20. Try these. = 2x + 3y • 4x + 3y -2x • 7x – 6y – 4x + 8y • 7z + 8x – 3z – 10x = 3x +2y =-2x +4z When we end up with a negative amount we can write a minus sign instead of a plus sign with a negative number.

21. What about exponents? X2 is like X∙X These are connected by multiplication and not addition or subtraction. This means that X2 is a single term. Here are some other examples of terms with multiple variables: X3, Y5, XY, XZ, XYZ, X2Y, X5Y3Z4

22. Try these examples. • 3x2 + 7xy + 5x2 + 3xy • 4x2 + 5x + 2x3 + 8x2 + 2x • 5xy + 3x + 4yx + 4y = 8x2 + 10xy Remember x, x2, and x3 are different terms. You only combine like terms. = 2x3 + 12x2 + 7x By the commutative property we can rearrange multiplication, so xy = yx, these are like terms. = 9xy +3x +4y

23. Still having trouble? • Try color coding, use colored pencils to identify the like terms. • Or use pictures for the situation (like worms). + 4xy + 6x + 2xy 4y 3x + + + + 6x 2xy 4y 3x + 4xy