Terms Handout TERMS Unit 2 Lesson 1 A. Identifying terms A very important skill in algebra is identifying the terms of an expression. You will notice that every term includes a sign, either positive or negative, except the first term when it is positive. Expressions Terms of the Expression 3x + 4 3x, +4 7y - 9 7y, -9-6x + 3y - 2-6x, +3y, -2-8 + 7-1 -8, +7, -1 4x 2-2x - 3y 3 + y - 7 4x 2, -2x, -3y 3, +y, -7 Exercises Identify the terms of each expression by circling them. Sample: Answer: 5 + 8x 5 + 8x 1. 3 + 7y 2. -14x - 6 3. 3x - 4y - 9 4. -5x 2-7x + 3 10. -9x - 4y + 3x - 2y - 7z 5. -7 + 8-2 6. 9-6 + 7-1 7. 5y - 7x + 6y + 8x + 2 8. 4x 2 + 6x - 7x + 3 9. -8y + 2 Terms Handout (pages 27 30) comes from IRSC s Academic Support Center. ws 01-4091 T.Lewis IRSC MAT0018 27
B. Identifying like terms Some terms have variables in them, like 2x or -3y 2 ; those are called variable terms. Other terms have no variables in them like -9 or 7; these are called constant terms or just constants. Like terms are those which have the same variables and exponents. Constant terms like -3, +7, and 0 are also like terms. In the expression 2x - 3y + 7x + 4y - 6, 2x and +7x are like terms and -3y and +4y are like terms. In the expression -3x 2 + 7x 2 + 4x - 3 + 8, -3x 2 and 7x 2 are like terms and -3 and + 8 are like terms; 7x 2 and 4x are not like terms; 4x and -3 are not like terms. Exercises Place circles around the first set of like terms you locate and rectangles around the next set. Sample: Answer: 5x + 6y - 7y - 9x 5x + 6y - 7y - 9x 1. -2x - 3x + y - 3y 2. 6x - 4y + 8x - 9y 3. 2x 2-3x + 7x 2 4. -7y - 3 + 8y + 1 5. -5y - 2y + 3x - 4y + 7x 6. 8x 2 + 7x 2 + 3-9 7. 14y - 3 + 8-9y 8. 4-17 - 3x - 9x 9. 9 + 3y - 7x + 18 10. -7x 2-4x 2 y - 9x 2 y + 10x 2 + 3x 2 y 11. 3-4x 12. 6x 2 + 8x 13. -6 + 7-5 + 5x T.Lewis IRSC MAT0018 28
C. Coefficient of a term The number factor of a variable term is called the coefficient. The coefficient of 3y is 3. The coefficient of -7x is -7. But what about the coefficient of a variable term like x or -x? The coefficient of x is 1 because x is understood to be 1x. The coefficient of -x is -1 because -x means the same thing as -1x. Exercises What is the coefficient of each variable term? Sample: -7x 2 Answer: -7 1. 2x 2 2. -5y 3. -7x 2 y 4. 8x 3 5. x 6. -5y 2 7. 7xyz 8. -z 9. xy 2 10. -3x T.Lewis IRSC MAT0018 29
D. Adding like terms The terms of an expression are its addends (quantifies that are added). Terms that are not alike may not be added. To combine like variable terms the coefficients are added; the variable and its exponent stay the same. To combine constant terms the constants themselves are added. Exercises Combine like terms. Sample: 2x + 3x Sample: -7-8 Answer: 5x Answer: -15 1. -3x + 5x 2. 7y - 2y + 11y 3. -6 + 8 4. 5-7 - 9 5. 4x - 8x + y + 6y 6. -3xy + 15xy - 7 + 2 7. 13x 2-2 + 7-20x 2 8. -6x - 3 + 4y + 17y - 9 + 15x 9. 3x 2-2xy + 7xy - 3 10. -16x 2-4 + 3x 2-4y + 7 T.Lewis IRSC MAT0018 30
Simplifying Algebraic Expressions Recall: Commutative Property of Addition changing the order of two addends does not change their sum Associative Property of Addition changing the grouping of addends does not change their sum Commutative Property of Multiplication changing the order of two factors does not change their product Associative Property of Multiplication changing the grouping of factors does not change their product Distributive Property of Multiplication over Addition the factor outside the parentheses is multiplied by each term in the sum Unit 2 Lesson 1 Example 1: Apply the commutative property of addition or multiplication to rewrite the expression. a) y + 5 b) g(2) c) -11 + s d) mn Example 2: Simplify each expression by combining like terms. a) 8m 14m b) 6a + a c) y 2 + 3y 2 + 7 d) 6z + 5 + z 4 e) 7y 5 + 8 + y f) -7y + 2 2y 9x + 12 x Example 3: Multiply. a) 6(4a) b) -8(9x) 31
c) 8(y + 2) d) 3(7a 5) e) 6(5 y) Example 4: Simplify. a) 5(2y 3) 8 b) -7(x 1) + 5(2x + 3) c) (y + 1) + 3y - 12 d) 8 6(w + 4) e) 20a 4(b + 3a) 5 f) 12 (5u + v) 4(u 6) + 2v 32
Vocabulary Solving Equations: The Addition and Multiplication Properties An equation is a mathematical sentence that uses an equal sign to state that two expressions represent the same number or are equivalent. Parts of an equation: Left side = Right side expressions Unit 2 Lesson 2 A solution of an equation is a value of the variable that makes the equation a true statement. 2x + 5 = 3(x 8) Differences between an Expression and an Equation Expression Equation An expression never contains an equal sign. An equation always contains an equal sign. We simplify an expression. We solve an equation. For example, we simplify 4x + 2x to get 6x. For example, we solve 4x + 2x = 6 to get x = 1 -vs- Example 1: Identify as an expression or an equation. Would you solve or simplify? a) 8 + 7b 9 b) -4(7x 9) c) -3(5n + 2) = 69 d) -7n 9n e) 1 2b = -8 5b Example 2: Determine whether the given number is a solution of the given equation. a) x = 5; 5 b) y 16 = -7; 9 c) 15 = -2g + 9; 3 d) 2(b 3) = 10; 1 33
The goal, when solving an equation, is to isolate the variable on one side of the equation (x = number or number = x). In other words, you want to get the variable x by itself on one side of the equation. Addition Property of Equality Let a, b, and c represent numbers. If a = b, then a + c = b + c In other words, the same number may be added to both sides of an equation without changing the solution of the equation. Example 3: Fill in the blank with the appropriate number. a) 11 + (-11) = b) 8 + = 0 c) + (-5) = 0 Example 4: Solve the equation using the addition property of equality. a) x 21 = 15 b) y 6 = -2 c) -1 = n 7 d) 4 = -17 + p e) x = -2 + 90 + (-100) 34
Subtraction Property of Equality Let a, b, and c represent numbers. If a = b, then a c = b c In other words, the same number may be subtracted from both sides of an equation without changing the solution of the equation. Example 5: Fill in the blank with the appropriate number. a) 50 = 0 b) 3 3 = c) 43 = 0 Example 6: Solve the equation using the subtraction property of equality. a) -2 = z + 8 b) 36 = y + 12 c) x + 12 = 4 d) 5 + c = 14 e) z = -28 + 36 35
Division Property of Equality Let a, b, and c represent numbers and let c 0. If a = b, then a b c c In other words, both sides of an equation may be divided by the same nonzero number without changing the solution of the equation. Example 7: Solve the equation using the division property of equality. a) 6y = 48 b) -2x = 26 c) -31 = -m d) -21 = -21w Multiplication Property of Equality Let a, b, and c represent numbers If a = b, then a c = b c In other words, both sides of an equation may be multiplied by the same nonzero number without changing the solution of the equation. Example 8: Solve the equation using the multiplication property of equality. a) n 5 11 b) x 9 9 c) v 0 8 d) w 8 6 36
Unit 2 Lesson 3 Solving Equations: Review of the Addition and Multiplication Properties When solving equations, you may have to combine like terms on each side of the equation before you use any of the properties of equality. Example 1: Solve each equation. First combine any like terms on each side of the equation. a) x + 6 = 1 3 b) 10 = 2m 4m c) -6y 1 + 7y = 17 + 2 d) -4 10 = 4y 5y e) z 20 22 4 37
When solving an equation, clear parentheses by using the distributive property if necessary. Example 2: Solve each equation. First multiply to remove parentheses. a) 13x = 4(3x 1) b) -2(-1 3y) = 7y c) -3 + 11 = -2z + 3(z 2) d) -15 5 = 9n 8(2 + n) Example 3: Solve each equation. a) 5y + 2 = 17 b) -11 = 3w 2 38
4a 4b 4c 4d 5b 5a 4 3 2b 2a 1 Are parentheses present? Use the distributive property to clear the parentheses. Look on the left side of your equation, are like terms present? Combine like terms. Look on the right side of your equation, are like terms present? Combine like terms. Are variable terms on the left and right side of your equation? Choose the smaller variable term then add or subtract that number on both sides. Are constant terms on the left and right side of your equation? Choose the constant term that s on the same side as the variable then add or subtract that number on both sides. Is there a number next to the variable? If the number is next to the variable divide that number on both sides. Is there a number underneath the variable? If the number is underneath the variable multiply that number on both sides. Example 4: Solve each equation. a) 8y 3 = 6y + 17 b) 6(z + 4) = -16 4z c) -4(x + 2) 60 = 2 10 d) -3(x + 9) 41 = 4 60 39
e) y 6 ( 1) 6 added to plus increased by more than total sum Addition Subtraction Multiplication Division rise subtract diminish multiply exceeds difference drop multiply by grow take away lose of larger less loss product than decreased lower times longer by minus twice/double than subtracted smaller enlarge from than gain less than Example 5: Translate each phrase into an algebraic expression. Let x be the unknown number. a) The product of 5 and a number, decrease by 25 divide quotient per divided by divided or shared equally b) Twice the sum of a number and 3 c) The quotient of 39 and twice a number d) Negative eight plus a number e) A number subtracted from twelve f) The difference of -9 times a number, and 1 40
1 2a 2b Solving Linear Equations in One Variable Are parentheses present? Use the distributive property to clear the parentheses. Look on the left side of your equation, are like terms present? Combine like terms. Look on the right side of your equation, are like terms present? Combine like terms. Unit 2 Lesson 4 1a 1b 1c 1d 5b 5a 4 3 Are variable terms on the left and right side of your equation? Choose the smaller variable term then add or subtract that number on both sides. Are constant terms on the left and right side of your equation? Choose the constant term that s on the same side as the variable then add or subtract that number on both sides. Is there a number next to the variable? If the number is next to the variable divide that number on both sides. Is there a number underneath the variable? If the number is underneath the variable multiply that number on both sides. Example 1: Solve each equation. a) 7x + 12 = 3x 4 b) 40 5y + 5 = -2y 10 4y c) 6(a 5) = 4a + 4 d) 26 (3x + 12) = 11 4(x + 1) 41
2a 2b 5b 5a 4 3 2b 2a 1 Are parentheses present? Use the distributive property to clear the parentheses. Look on the left side of your equation, are like terms present? Combine like terms. Look on the right side of your equation, are like terms present? Combine like terms. Are variable terms on the left and right side of your equation? Choose the smaller variable term then add or subtract that number on both sides. Are constant terms on the left and right side of your equation? Choose the constant term that s on the same side as the variable then add or subtract that number on both sides. Is there a number next to the variable? If the number is next to the variable divide that number on both sides. Is there a number underneath the variable? If the number is underneath the variable multiply that number on both sides. Example 2: Solve each equation. a) -4 + 2x + 1 = 3(x 1) b) 2(2v + 3) + 8v = 6(v 1) + 3v Words that represents the equal sign Equals Gives Is/was Yields Amounts to Is equal to Example 3: Translate each sentence into an equation. a) The difference of 110 and 80 is 30. b) The produce of 3 and the sum of -9 and 11 amounts to 6. c) The quotient of 24 and -6 yields -4. 42
added to plus increased by more than total sum Unit 2 Lesson 4 Problem Solving Addition Subtraction Multiplication Division Equal Sign minus multiply divide equals difference multiply by quotient gives less of per is/was decreased by product divided by yields subtracted from times divided or amounts less than twice/double shared to equally is equal to Example 1: Write each sentence as an equation. Use x to represent a number. a) Four times a number is 20. b) The sum of a number and -5 yields 32. c) Fifteen subtracted from a number amounts to -23. d) Five times the difference of a number and 7 is equal to -8. e) The quotient of triple a number and 5 gives 1. Example 2: Translate each sentence into an equation. Then solve the equation. a) The sum of a number and 2 equals 6 b) Five times the sum of a number and 2 added to three times the number. is 11 less than the number times 8. Find the number. Find the number. 43
Example 3: a) Terry ate twice as many Skittles as her friend Karen. If Karen ate x Skittles, write an expressions for the number that Terry are. b) One week Kerry worked 9 hours more that her friend Robert. If Robert worked x hours, write and expression for the number of hours that Kerry worked. c) Dorian s daughter is 30 years younger than he is. Write an expression for his daughter s age if Dorian is A years old. Example 4: Solve. a) The distance by road from Cincinnati, Ohio, to Denver, Colorado, is 71 miles less than the distance from Denver to San Francisco, California. If the total of these two distances is 2399 miles, find the distance from Denver to San Francisco. b) A woman s $57,000 estate is to be divided so that her husband receives twice as much as her son. How much will each receive? 44