Transcript Simplifying Expressions with Real Numbers

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.
1
5
5x  x  7 , 4 ,
xy  ,
3
7
2
 7 x  2 
Consider the example: 5 x  x  7
2
The terms of the expression are separated by addition.
There are 3 terms in this example and they are
5x 2 , x ,  7 .
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
To simplify some expressions we may
need to use the Distributive Property
Do you remember it?
Distributive Property
a ( b + c ) = ba + ca
Examples
Example 1: 6(x + 2)
Distribute the 6.
Example 2: -4(x – 3)
Distribute the –4.
6 (x + 2) = x(6) + 2(6)
= 6x + 12
-4 (x – 3) = x(-4) –3(-4)
= -4x + 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)
Example 4: -(4x – 3)
= 1( x – 2 )
= -1(4x – 3)
= x(1) – 2(1)
= 4x(-1) – 3(-1)
=x - 2
= -4x + 3
Notice multiplying by a 1 does
nothing to the expression in the
( )’s.
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
Unlike Terms
5x , 8y
-6.7xy , 02xy
 3x y , 8xy
The variable factors are
identical.
2
2
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
4 x 2  13 y  4 x  12 x 2  3 x  3
Reorder the terms.
4 x 2  12 x 2  4 x  3 x  13 y  3
Combine like terms.
16 x 2  x  13 y  3
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
1
6 x  10  3x  4
2
Simplifying Example
Distribute.
1
6 x  10  3x  4
2
Simplifying Example
Distribute.
1
6 x  10  3x  4
2
1
1
6 x   10   x3  43
2
2
3 x  5  3 x  12
Simplifying Example
Distribute.
1
6 x  10  3x  4
2
1
1
6 x   10   x3  43
2
2
3 x  5  3 x  12
Combine like terms.
Simplifying Example
Distribute.
1
6 x  10  3x  4
2
1
1
6 x   10   x3  43
2
2
3 x  5  3 x  12
Combine like terms.
6x  7
Evaluating Expressions
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.
Remember to use correct order of operations.
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
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Evaluating Example
Evaluate x 2  4 xy  3 y 2 when x  2 and y  1
Evaluating Example
Evaluate x  4 xy  3 y when x  2 and y  1
2
2
Substitute in the numbers.
Evaluating Example
Evaluate x  4 xy  3 y when x  2 and y  1
2
2
Substitute in the numbers.
22  42 1  3 12
Evaluating Example
Evaluate x 2  4 xy  3 y 2 when x  2 and y  1
Substitute in the numbers.
22  42 1  3 12
Remember correct order of operations.
4  421  31
483
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Common Mistakes
Incorrect
Correct