Basic Algebra/Working with Numbers/Combining Like Terms

Lesson
Algebra is used to make many problems simpler, and that is why a lot of algebra is about finding simple expressions which mean the same thing as harder ones. Variables are given different letters and symbols in algebra so they can be kept apart, so every time $$x$$ is used in an expression it means the same thing, and every time $$y$$ is used it means the same thing, but a different thing to $$x$$ (of course this is only in the same expression, different expressions can use the same letters to mean different things). Since the different letters keep the variables apart this means that an expression with many variables in many places can be made simpler by bringing them together.

Example Problems
Here is an example of variables keeping numbers apart even if we don't know them, and this lets us combine them without changing their value: Albert has some books in his bag, he does not know how many. Beth also has some books and she does not know how many. Chris does not know how many books he has, but he knows it is the same as Beth. Dora knows she has the same number of books as Albert. In this example there are 4 lots of books, so we could write the total number of books as:

$$a + b + c + d$$

Since we know that Albert and Dora have the same number of books, and Chris and Beth have the same number of books, we could also write:

$$a + b + a + b$$

This is the same as writing:

$$2a + 2b$$

Here we have grouped both a terms and both b terms. We could also go further, since everything is being multiplied by 2, and write:

$$2(a + b)$$

This is the simplest way of writing how many books there are. Not only were the variables combined, but so were the constants (in this case the number 2). We can check if they are the same by seeing what happens when Albert has 2 books and Beth has 5.

$$a + b + c + d = 2 + 5 + 2 + 5 = 14$$

$$2(a + b) = 2(2 + 5) = 2 \times 7 = 14$$

Practice Games
Put links here to games that reinforce these skills.


 * http://www.math.com/school/subject2/practice/S2U2L4/S2U2L4Pract.html
 * http://www.studystack.com/matching-47007

Practice Problems
 {Simplify these into the form $$x(y + z)$$ where $$x$$, $$y$$ and $$z$$ are integers or variables.}

{ $$15a + 19b + 2a - 2b =$${ 17(a + b) (i)|17(a+b) (i) _10 }
 * type="{}"}

{ $$a + 9 + 2a =$${ 3(a + 3) (i)|3(a+3) (i) _10 }
 * type="{}"}

{ $$12a + 16b =$${ 4(3a + 4b) (i)|4(3a+4b) (i) _10 }
 * type="{}"}

{ $$3a - 3b =$${ 3(a - b) (i)|3(a-b) (i) _10 }
 * type="{}"}

{ $$5a + a^2 =$${ a(5 + a) (i)|a(a + 5) (i)|a(5+a) (i)|a(a+5) (i) _10 }
 * type="{}"}