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Understanding Algebra: Combining Like Terms and Identifying Terms

This section covers the basics of algebra, focusing on identifying terms, variables, coefficients, and constants. It explains that variables represent unknown numbers and that expressions consist of numbers, variables, and operation symbols. The chapter details how to identify like terms—terms that share the same variables and exponents—and teaches how to combine them by adding or subtracting their coefficients. Additionally, it touches on the distributive property and how to simplify expressions by removing parentheses, ensuring a solid foundation in algebraic principles.

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Understanding Algebra: Combining Like Terms and Identifying Terms

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  1. MTH 10905Algebra Combining Like Terms CHAPTER 2 Section 1

  2. 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

  3. Terms • Terms are the parts that are added together. • EXP: 5x – 8y – 2 EXP: 2(x – 4) – 7x + 3 5x + (-8y) + (-2) 2(x - 4) + (-7x) + 3 3 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.

  4. Numerical Coefficient • Numerical CoefficientorCoefficientis 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.

  5. Identify Terms • 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.

  6. Identify Terms • Like Terms EXP: 3x and -6x 2y and 8y 2x2 and -3x2 4ab and 5ab 3 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

  7. 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. • 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

  8. 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:

  9. 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

  10. 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?

  11. 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?

  12. 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

  13. 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) Coefficient is assumed to be 1(2x + 4) (1)(2x) + (1)(4) 2x + 4 EXP: (x + 5) x + 5

  14. 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

  15. Simplify an Expression • Simplifying an expresstion • Use the distributive property to remove the parentheses • Combine like terms EXP: 8 – (3x + 7) 8 + (-3x) + (-7) 8 – 3x – 7 -3x + 8 – 7 -3x +1

  16. 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

  17. Simplify an Expression EXP:

  18. Simplify an Expression EXP:

  19. HOMEWORK 2.1 • Page 103 – 104 #9, 11, 18, 21, 47, 59, 64, 74, 79, 93, 97, 117, 121

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