Simplifying Expressions
By: Karen Overman
Objective
This presentation is designed to give a brief review of simplifying algebraic expressions and evaluating algebraic expressions.
Algebraic Expressions
An algebraic expression is a collection of real numbers, variables, grouping symbols and operation symbols.
Here are some examples of algebraic expressions.
Consider the example:
The terms of the expression are separated by addition. There are 3 terms in this example and they are .
The coefficient of a variable term is the real number factor. The first term has coefficient of 5. The second term has an unwritten coefficient of 1.
The last term , -7, is called a constant since there is no variable in the term.
Let’s begin with a review of two important skills for simplifying expression, using the Distributive Property and combining like terms. Then we will use both skills in the same simplifying problem.
Distributive Property
a ( b + c ) = ba + ca
To simplify some expressions we may need to use the Distributive Property
Do you remember it?
Distributive Property
Examples
Example 2: -4(x – 3)
Distribute the –4.
-4 (x – 3) = x(-4) –3(-4)
= -4x + 12
Example 1: 6(x + 2)
Distribute the 6.
6 (x + 2) = x(6) + 2(6)
= 6x + 12
Practice Problem
Try the Distributive Property on -7 ( x – 2 ) .
Be sure to multiply each term by a –7.
-7 ( x – 2 ) = x(-7) – 2(-7)
= -7x + 14
Notice when a negative is distributed all the signs of the terms in the ( )’s change.
Examples with 1 and –1.
Example 3: (x – 2)
= 1( x – 2 )
= x(1) – 2(1)
= x - 2
Notice multiplying by a 1 does nothing to the expression in the ( )’s.
Example 4: -(4x – 3)
= -1(4x – 3)
= 4x(-1) – 3(-1)
= -4x + 3
Notice that multiplying by a –1 changes the signs of each term in the ( )’s.
Like Terms
Like terms are terms with the same variables raised to the same power.
Hint: The idea is that the variable part of the terms must be identical for them to be like terms.
Examples
Like Terms
5x , -14x
-6.7xy , 02xy
The variable factors are
identical.
Unlike Terms
5x , 8y
The variable factors are
not identical.
Combining Like Terms
Recall the Distributive Property
a (b + c) = b(a) +c(a)
To see how like terms are combined use the
Distributive Property in reverse.
5x + 7x = x (5 + 7)
= x (12)
= 12x
Example
All that work is not necessary every time.
Simply identify the like terms and add their
coefficients.
4x + 7y – x + 5y = 4x – x + 7y +5y
= 3x + 12y
Collecting Like Terms Example
Both Skills
This example requires both the Distributive
Property and combining like terms.
5(x – 2) –3(2x – 7)
Distribute the 5 and the –3.
x(5) - 2(5) + 2x(-3) - 7(-3)
5x – 10 – 6x + 21
Combine like terms.
- x+11
Simplifying Example
Simplifying Example
Distribute.
Simplifying Example
Distribute.
Simplifying Example
Distribute.
Combine like terms.
Simplifying Example
Distribute.
Combine like terms.
Evaluating Expressions
Remember to use correct order of operations.
Evaluate the expression 2x – 3xy +4y when
x = 3 and y = -5.
To find the numerical value of the expression, simply replace the variables in the expression with the appropriate number.
Example
Evaluate 2x–3xy +4y when x = 3 and y = -5.
Substitute in the numbers.
2(3) – 3(3)(-5) + 4(-5)
Use correct order of operations.
6 + 45 – 20
51 – 20
31
Evaluating Example
Evaluating Example
Substitute in the numbers.
Evaluating Example
Substitute in the numbers.
Evaluating Example
Remember correct order of operations.
Substitute in the numbers.
Common Mistakes
Incorrect
Correct