MTH 10905Algebra

COMBINING LIKE TERMS

CHAPTER 2 SECTION 1

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

Identify Terms

Terms are the parts that are added together.

EXP: 5x – 8y – 2 EXP: 2(x – 4) – 7x + 35x + (-8y) + (-2) 2(x - 4) + (-7x) + 33 terms: 3 terms:

5x , -8y , -2 2(x - 4) , -7x , 3

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.

Terms

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.

Numerical Coefficient

When a variable has no coefficient we assume it is 1. EXP: x = 1x EXP: x2 = 1x2

EXP: (x + 3) = 1(x + 3) 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 2y and 8y2x2 and -3x2 4ab and 5ab3 and 4 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

Identify 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. Determine which terms are alike2. Add or subtract the coefficients of the like terms3. Multiply the number found in step 2 by the common

variables

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:

Combine Like Terms

2 4 2 5 10 4 3 12 LCD = 15 and 3 5 3 5 15 5 3 15

10 12 10 12 215 15 15 15 15

x x

x x x x

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

Combine Like Terms

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:

Can the distributive property be used when we have 3 terms?

Distributive Property

2 2 2 10 18 10(5 9) (5 ) (9) 63 3 3 3 3 3t t t t

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

Distributive Property

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) Coefficient is assumed to be 1(2x + 4)(1)(2x) + (1)(4)2x + 4

EXP: (x + 5)x + 5

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

Parentheses

Simplifying an expresstion1. 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) –

8x30x – 18 – 2y + 8 – 8x30x – 8x – 2y – 18 + 822x – 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:

Simplify an Expression

5 1 (3 1)6 4

5 1 13 ( 1)6 4 4

5 3 1 5 2 10 3 3 9 LDC = 12 and 6 4 4 6 2 12 4 3 12

10 9 112 12 4

19 112 4

x x

x x

x x

x x

x

EXP:

Simplify an Expression

4 25 3

4 25 3

4 25 3

4 215 3

4 5 25 5 3

1 25 3

x x

x x

x x

x x

x x

x

HOMEWORK 2.1

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