Basic Number Properties Unit Five
Using Properties to simplify Expressions and Solve Equations
So how do all these properties help us anyway?
In this unit we will apply all we have learned to solving algebraic equations and simplifying algebraic expressions.
Before we can start the unit we have to talk about the difference between an expression and an equation.
Basically a mathematical expression has no = sign but can be simplified.
The simplification may or may not have a value.
Consider the following expressions
a * 1/a has a value of 1
a(2 + 3) can be simplified to 5*a or 5a but does not have a specific value
An equation is an expression that equals something
a * 2 = 4
6a + 10 = 70
Equations can be solved so that the value of a variable can be found.
Simplifying an Algebraic Expression
Consider the following expression: (-3/7)(7-42a)
| Property | Result | |
|---|---|---|
| Apply the distributive property | (-3/7 * 7) - (-3/7 * 42a) | |
| Apply definition of Multiplication | (-3) - (-18a) | |
| Apply definition of subtraction | (-3) + 18a | |
(Notice that -3 * 42 / 7 = 18)
Your expression is simplified! (Why can't we add -3 to 18a?)
Lets do another one: 27rst + 27tsr
| Property | Result | |
|---|---|---|
| Apply commutative property of multiplication | 27rst + 27rst | |
| Apply Definition of Addition | 54rst | |
Your expression is simplified!
Try this! 5 + h(6/h)
| Property | Result | |
|---|---|---|
| Apply Associative property of multiplication | 5 + 6* h/h | |
| Apply definition of Division | 5 + 6 * (h * 1/h) | |
| Apply inverse property of multiplication | 5 + 6 * 1 | |
| Apply identity property of multiplication | 5 + 6 | |
| Apply definition of Addition | 11 | |
Using Properties to Solve Equations
Before you can solve any equation it is important to simplify the expressions on either side of the equal sign.
Consider : 4n - 2n = 18
| Property | Result | |
|---|---|---|
| Applying Definition of Addition we can simplify 4n - 2n | 2n = 18 | |
We have simplified for n but we do not have a value yet.
Remember we can manipulate our equation anyway we want as long as we....
How do we get n by itself?
We have to eliminate the 2 so instead of 2n we just have n.
What two properties will help us?
If you said Identity and Inverse property of multiplication you are correct!
Watch: 2n = 18 *** Remember to treat each side equally! ***
| Property | Result | |
|---|---|---|
| Applying the Inverse property of multiplication | 1/2 * 2n = 18 * 1/2 | |
| Applying the Definition of multiplication | 1n = 9 | |
| Applying the Identity property of multiplication | n = 9 | |
9 is the value of n!
Lets see, if we substitute the value of 9 for n is our equation true?
4(9) - 2(9) = 18
- 36 - 18 = 18
- 18 = 18
Yes, we are correct!!
Let's do one more before you try it on your own!
2x - 4 = 10
Notice that we can't simplify 2x - 4 because they are not like terms.
So now we just solve by manipulating the equation until x is by itself. But remember...
| Property | Result | |
|---|---|---|
| Applying the Inverse property of addition | 2x - 4 (+4) = 10 (+4) | |
| Applying the Definition of addition | 2x + 0 = 14 | |
| Applying the Identity property of addition | 2x = 14 | |
| Applying the Inverse property of multiplication | 1/2 * 2x = 14 * 1/2 | |
| Applying the Definition of multiplication | 1x = 7 | |
| Applying the Identity property of multiplication | x = 7 | |
The value of x is 7.
Let's see if we are correct by substituting the value of 7 for x in the original equation.
2(7) - 4 = 10
14 - 4 = 10
- 10 = 10
Quick Check
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Quick check before you proceed.
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