Download Combining Like Terms

MTH 10905
Algebra
COMBINING LIKE TERMS
CHAPTER 2 SECTION 1
Identify Terms
 Variables are the letters or symbols that represent numbers and are
used when different numbers can be used in a situation.

EXP: x, y, z
EXP: ☼ , ☺ , ♥
 Expression or Algebraic Expression is the collection of numbers,
variables, grouping symbols, and operation symbols.

EXP: 5
EXP: 2x – 3y – 5

EXP: x2 – 3
EXP: 3(x + 7) + 2

EXP: (x + 3)
4
Terms
 Terms are the parts that are added together.
5x – 8y – 2
5x + (-8y) + (-2)
3 terms:
5x , -8y , -2
EXP: 2(x – 4) – 7x + 3
2(x - 4) + (-7x) + 3
3 terms:
2(x - 4) , -7x , 3

EXP:

It is important that you list the minus sign when identifying a
term.

In the Language of Mathematics we often assume that readers
know certain things by the absence of a symbol. Example:
5x is assumed to have a positive sign associated with it because
no sign is given.
Numerical Coefficient
 Numerical Coefficient or Coefficient is the number part of the
term.

EXP:
6x
6 is the coefficient

EXP:
5(x – 2)
5 is the coefficient
 The variable is multiplied by the coefficient
 Variables are used when different numbers can be used in a
situation.
Identify Terms
 When a variable has no coefficient we assume it is 1.
EXP:
 EXP:

x = 1x
(x + 3) = 1(x + 3)
EXP: x2 = 1x2
EXP: ab = 1ab
 Constant term or constant is the term that has no variable and are
used when only one number can be used in a situation.
 EXP: If you are charged a monthly fee of $9.95 for internet
service and an hourly fee of $1.25. You charges are represented by
a constant $9.95 and a variable $1.25x.
1.25x + 9.95
 Like Terms or Similar Term are terms that have the same
variables with the same exponents. Constants are also like terms.
Identify Terms
 Like Terms
EXP:
3x and -6x
2x2 and -3x2
3 and 4
2y and 8y
4ab and 5ab
2(x + 1) and -6(x + 1)
 Identify any like terms
EXP: 6a2 + 7a + 5a2
Like terms: 6a2 and 5a2
EXP: 9 – 3x + 8x – 11
Like terms: 9 and -11 ; -3x and 8x
Like Terms
 Identify Like Terms:
EXP: 6a + 7b + 5
Like terms: None
EXP: 9 – 3x + 8x – 11
Like terms: 9 and -11 ; -3x and 8x
 Combine Like Terms means to add or subtract the like terms in an
expression.
1.
2.
3.
Determine which terms are alike
Add or subtract the coefficients of the like terms
Multiply the number found in step 2 by the common
variables
Combine Like Terms
 EXP: 6x + 7x = 13x
 To make math expression more real for you replacement the
variable with a word such as cookies. Then you would have
6 cookies + 7 cookies = 13 cookies
 You can also relate the addition to the distributive property.
6x + 7x = (6 + 7)x
 EXP:
2
4
x x 
3
5
LCD = 15
2 5 10
 
3 5 15
10
12
2
 10 12 
x x   x  x
15
15
15
 15 15 
and
4 3 12
 
5 3 15
Combine Like Terms
 EXP:
3.72a – 8.12a = (3.72 – 8.12)a = 4.40a
 EXP:
2x + x + 10 = 2x + 1x + 10 = (2 + 1)x + 10 = 3x + 10
 When writing your answers we generally list the terms that contain
variables in alphabetical order from left to right, and the constant
as the last term.
 The commutative property, a + b = b + a, and the associative
property, (a + b) + c = a + (b + c), of addition allows us to
rearrange the terms.
 EXP:
7b + 9c – 12 + 5c = 7b + 9c + 5c – 1 2
7b + (9 + 5)c – 12 = 7b + 14c – 12
Distributive Property
 EXP:
-5x2 + 7y – 3x2 – 9 – 2y + 4
-5x2 – 3x2 + 7y – 2y – 9 + 4
(-5 + -3) x2 + (7 – 2)y + (-9 + 4)
-8x2 + 5y – 5
 Understanding Subtraction of Real Number will help us understand
the use of the distributive property .
a – b = a + (-b)
 Distributive property is used to remove parentheses
a(b + c) = ab + ac
How can you simplify using the distributive property?
Distributive Property
EXP:
6(x + 2) = 6x + (6)(2) = 6x + 12
EXP:
-4(r + 3) = -4r + (-4)(3) = -4x – 12
EXP:
8(w – 7) = 8w + (8)(-7) = 8w – 56
EXP:
-3(r – 9) = -3r + (-3)(-9) = -3r + 27
EXP:
2
10 18
10
 2
 2
 (5t  9)     (5t )     (9)   t    t  6
3
3
3
3
 3
 3
Can the distributive property be used when we have 3 terms?
Distributive Property
 Remember that the distributive property can be expanded to more
than two terms.
EXP:
-4(2x + 3y -5z)
(-4)(2x) + (-4)(3y) + (-4)(-5z)
-8x – 12y + 20z
 The distributive property can also be used from the right.
EXP:
(3a – 4b)2
(2)(3a) + (2)(-4b)
6a – 8b
Parentheses
 Removing parentheses when they are preceded by a plus or minus
sign using the distributive property
 When no sign or plus sign before the parentheses we simply
remove the parentheses.
EXP:
(2x + 4)
1(2x + 4)
(1)(2x) + (1)(4)
2x + 4
EXP:
(x + 5)
x+5
Coefficient is assumed to be
Parentheses
 When a minus sign comes before the parentheses, all of the signs
within the parentheses changes
EXP:
-(3x + 2)
-1(3x + 2)
(-1)(3x) + (-1)(2)
-3x – 2
EXP:
-(x + 4)
-x – 5
Simplify an Expression

Simplifying an expresstion
1. Use the distributive property to remove the parentheses
2. Combine like terms
EXP:
8 – (3x + 7)
8 + (-3x) + (-7)
8 – 3x – 7
-3x + 8 – 7
-3x +1
Simplify an Expression
EXP:
6(5x – 3) – 2(y – 4) – 8x
(6)(5x) + (6)(-3) + (-2y) + (-2)(-4) – 8x
30x – 18 – 2y + 8 – 8x
30x – 8x – 2y – 18 + 8
22x – 2y – 10
Remember there is more than one way to write a
solution. However if we all use the same style it
helps us compare our solutions
Simplify an Expression
EXP:
5
1
x  (3x  1)
6
4
5
1
1
x     3x     (1)
6
4
4
5 x 3x 1
 
6
4 4
10
9
1
x x
12
12
4
19
1
x
12
4
LDC = 12
5 2 10
 
6 2 12
and
3 3 9
 
4 3 12
Simplify an Expression
EXP:
2
4
 x    x
3
5
4
2
 x x
5
3
4
2
 xx
5
3
4
2
 x  1x 
5
3
4
5
2
 x x
5
5
3
1
2
x
5
3
HOMEWORK 2.1
 Page 103 – 104
#9, 11, 18, 21, 47, 59, 64, 74, 79, 93, 97, 117, 121