1. Algebra

2. Expression and Equation
Expression: An Expression can be written using a single constant, a single variable, or a combination of operations with constant, variable, or numerical coefficients. For example 2y – 7 constant numerical coefficient variable Equation: A Mathematical statement with two expression that have same value. x + 2 = 3 y – 7 = - 4 3a – 2 = a + 2 and b = 4 are examples of equation

3. Properties
Properties are set of math rules that are always true. They often help us solve equations. Here are some important ones.

4. Identity Property
Identity Property of Addition looks like this a + 0 = a. It says that if you add zero to any number, that number stay the same. Example : = 5
Identity Property of Multiplication looks like this a * 1 = a. It says that if you multiply to any number by 1, that number stay the same. Example : * 1 = 7

5. Associative Property
The Associative Property of Addition look likes this: (a + b) + c = a + (b + c). It says that when adding three different numbers, you can change the order that you add them by moving the parentheses and the answer will still be the same.
Example: (2+5)+8 = 2+(5+8) (both expression equal 15)

6. Distributive Property
The Distributive Property of Multiplication Over Addition look likes this: a(b + c) = ab + ac It says that adding two numbers inside parentheses, then multiplying that sum by a number outside the parentheses is equal to first multiplying the number outside the parentheses by each of the numbers inside the parentheses and then adding the two product together.

7. Distributive Property
The distributive property allows us to simplify an expression by taking out the parentheses. Example: 2(4+6) = 2 * * 6 (you “distribute” the “2 *” across the term inside the parentheses. Both expression equal 20) Example: 7(y+8) (y + 8) = 7(y) + 7(8) = 7y + 56
Think about catapulting the number outside the parentheses inside to simplify

8. Distributive Property
The Distributive Property of Multiplication Over Subtraction look likes this: a(b - c) = ab - ac It says that subtracting two numbers inside parentheses, then multiplying that difference times a number outside the parentheses is equal to first multiplying the number outside the parentheses by each of the numbers inside the parentheses and then subtracting the two products.

9. Distributive Property
The Distributive Property of Multiplication Over Subtraction Example 1 : 9(5 – 3) = 9(5) – 9(3) (Both expression equal 18) Example 2 : 6(y – 8) (y – 8) = 6(y) – 6(8) = 6y - 48

11. Factoring
FACTORING is the reverse of the distributive property. Instead of getting rid of parentheses, factoring allows us to include parentheses (because sometimes it’s simpler to work with an expression that has parentheses)

12. Example: Factor 15y + 12 Step 1: Ask yourself “what is the greatest common factor of both term?” In this case the GCF of 15y and 12 is 3. (15y = 3*5*y and 12 = 3*4) Step 2: Divide all term by the GCF and put the GCF on the outside of the parentheses. 15y + 12 = 3(5y + 4) ** Note: You can always check your answer by using the DISTRIBUTIVE PROPERTY. Your answer should match the expression you started with!

13. Balance as a model of an equation

14. In the pictures below, each circle represents one and the block represents the unknown x. To find out what the block weighs, you can add the same amount (circles or blocks) to BOTH sides take away the same amount from BOTH sides. That way both sides will maintain the balance or "the equality".
x + 3 = 5
If this is a balanced situation...  
x = 2
...so is this! (We took away three circles from BOTH sides.)

15. Balance Equations
3x + 2 = 2x + 6  
Take away two blocks (two x's) from both sides. The balance will stay balanced.
x + 2 = 6  
Take away 2 circles from both sides. The balance will stay balanced.
x = 4
Here is the solution!

16. Without the scale model, the solving process looks like this:
3x + 2 = 2x x - 2x (take away 2x from both sides) x + 2 = (take away 2 from both sides) x = 4
                       

17. Dividing
2x = 8  
If you take away half of the things on the left side, and similarly half of the things on the right side, the balance will stay balanced.
x = 4

18. 3x = 9  
Think about it! If one is a balanced situation...
x = 3
...so is the other (and vice versa)! We simply divided both sides by 3.

19. Subtracting and Dividing
4x + 2 = 2x + 5
First we get rid of the blocks on the right side by taking away two blocks from both sides.
2x + 2 = 5
Then we eliminate the circles on the left side by removing 2 circles from both sides.
2x = 3
Now there are only blocks on one side and only circles on the other. To find out what 1 block weighs, we take half of both sides.
x = 1 1/2
The solution is that 1 block weighs 1 1/2 circles.

20. Practice 2x + 3 = 5 2x + 5 = x + 9 3x + 2 = 2x + 4 3x + 3 = 5 + x

21. Using the balance model with negative terms in an equation
It is not as easy to portray negative numbers in an equation with the balance model, because in nature, we don't have "negative weight." But there is a way: you can think of the negatives as holes or empty spots that, when filled with positives, become "nothing" or zero.

22. Solve: x − 4 = 3
x − 4 = 3
Add 4 circles to both sides to "fill in" the negative circles.
x − = 7
Now we have added 4 circles to both sides of the balance/equation.
x = 7
The 4 circles "fill in" or cancel the negatives. So in the end there is nothing − or zero − left.

23. Solve 2x + 3 = −5.
2x + 3 = −5
We want the left side to have blocks ONLY (the unknowns). Therefore we have to eliminate three circles from both sides.
2x = −8
But since there are no circles to take away on the right side, we have to "add" negative circles.
x = −4
As a last step, we take half of each side to arrive to the solution x = −4.

24. 2x − 3 = −x + 3 To isolate x (to have it alone) on the left side, we need to eliminate −x on the right side. Since it is negative, we ADD x to both sides. 2x − 3 + x = −x x Now −x and x cancel each other (on the right side). 2x − 3 + x = 3 Then we can add 2x and x on the left side to get 3x. 3x − 3 = 3 The next step is to eliminate −3 on the left side. For that end we add 3 to both sides. 3x − = Again, −3 and + 3 cancel each other 3x = 6 As the last step, since there are three x's on the left side and we want to know how much one is worth, we divide both sides of the equation by three. x = 2 This is the final solution

25. It checks, so the solution was correct.
To check the solution, we substitute x = −4 to the equation 2x + 3 = −5:
2( -4 ) + 3 = 5
= 5
-5 = -5
It checks, so the solution was correct.

26. Practice x − 5 = 5 2x − 5 = 5 3x − 4 = 2x + 4 5x − 3 = −x + 3