CHAPTER 10: ONE-STEP ALGEBRAIC EQUATIONS SIMPLIFYING ALGEBRAIC EXPRESSIONS p WRITING ALGEBRAIC EXPRESSIONS p 4-5

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1 CHAPTER 10: ONE-STEP ALGEBRAIC EQUATIONS 10.1 SIMPLIFYING ALGEBRAIC EXPRESSIONS p WRITING ALGEBRAIC EXPRESSIONS p EVALUATING ALGEBRAIC EXPRESSIONS p ONE-STEP ADDITION AND SUBTRACTION EQUATIONS p ONE-STEP MULTIPLICATION EQUATIONS p ONE-STEP DIVISION EQUATIONS p ONE-STEP EQUATIONS WITH FRACTION COEFFICIENTS p ANSWER KEY P

2 10.1 SIMPLIFYING ALGEBRAIC EXPRESSIONS Key Vocabulary 10.1 algebraic expression: variable: coefficient: constant term: like terms: unlike terms: Think 10.1 coefficient 2x 2 exp onent var iable Like terms have same variables raised to same exponent y and 3xy share the like term y. 2x5 3 share the like terms 5 and 3. Unlike terms do not share a common variab le raised to same exponent. 2 2x and 2x do not share a common variable mn and 2m n do not share a common variable. Constant terms stand alone without exponent. For 2x 3,3is a constant term because it has no variable. You can simplify algebraic expressions by combining like terms only. 2x x 3x 3xy 2xy 5xy Try It 10.1 Identify the parts of the algebraic expression. Coefficient Variable Exponent 1) 2y 2 2) 3x 3 Circle and combine the like terms. 3) 5x + 3x 4= 4) xy 4y + 3xy= 5) 2x 2 + 7y x 2 + y= 6) 4m + 2n m + 3n= 2

3 10.1 PRACTICE EXERCISES Identify the parts of the terms. Coefficient Variable Exponent 1) 3x 2 2) 6xy 2 3) 3x + 2y 2 Circle the like terms. 4) 7x y 2 + x + xy 5) 2x 3 + x 2 + 2y + 3x 2 6) 4m n + 2m 2 + 6n Simplify by combining like terms. 7) 3x 2y 2 + x = 8) 2x 3 + 2x 2 + 3x 3 = 9) 4m 3 + m 2 + m 2m 3 = 10) 3x 2 + 4x x 2 x = 3

4 10.2 Writing Algebraic Expressions Key Vocabulary 10.2 word expression: Think 10.2 Operation Algebraic Expression Word Expression Addition Subtraction 4 + y x m m 4 The sum of 4 and y 7 added to a number x 6 decreased by a number m 4 less than a number m Multiplication 6x The product of 6 and a number x Division 3 n 3 divided by n n The quotient of a number divided by 4 decrease by 7 Try It 10.2 Algebraic Expression More examples: Five more than three time a number is written as 3n + 5 The product of a number and 5 is written as 5n Six divided by twice a number is written as 6 2n or 6 2n Four times a number decreased by 11 is written as 4n - 11 Word Expression 1) 6 + m 2) 2y 10 3) 8 + x 2 Word Expression Algebraic Expression 4) he product of a number and 5 5) n divided by 2 6) five more than three time a number 4

5 10.2 PRACTICE EXERCISES Write as an algebraic expression. 1) y increased by 5: 2) The product of n times the number 2: 3) The product of 3 times a number squared: 4) Fifteen less than twice a number: 5) The product of nine and a number, decreased by six: Write as a word expression. 6) 3n + 7: m 7) 6 : 2 8) Tasha is n inches tall. Her mother 15.6 centimeters taller. Write an algebraic expression to show her mother s height. 9) Some friends (f) are going to share equally the cost of dinner, Dinner was $ total. Write an algebraic expression to show how much each person will share. 10) Carol bought some chairs (c). The cost for each chair was $ Write an algebraic expression to show the total cost for the chairs. 5

6 10.3 EVALUATING ALGEBRAIC EXPRESSIONS Key Vocabulary 10.3 evaluate: Think 10.3 First substitute the given number for the variables. Then use PEMDAS rules for order of operations to find the values of the algebraic expressions y + 7, when y = = 10 6x, when x = 4 6 ( 4) = m, when m = x + y, when x = 2 and y = = = x 10, when x Try It 10.3 Evaluate the algebraic expression. 1) 2x + 3, when x = 5 = 2) 3y + x, when x = 5 and y = 3 = 3) y 1, when y = 24 = ) 2xy, when x = 3 and y = 4 = 5) 3a 2, when a = 3 = 6) mn, when m = 5 and n = 10 = 2 6

7 10.3 PRACTICE EXERCISES Evaluate the algebraic expressions. 1) x 2 + y 2 : x = 4, y = 5 2) 3xy: x = 3, y = 5 3) 4x 2y: x = 6, y = 3 4) 3 + (ab) 2 : a = 3, b = 4 m 5) 10 : m = ) x: x = 4 7) $10 + $3y: y = $10 8) 16 2m n : n = 4, m = 3 9) 2a + 2b + 2c: a = 3, b = 4, c = 5 10) 3xyz: x = 3, y = 2, z = 1 7

8 10.4 One Step Addition and Subtraction Equations Key Vocabulary 10.4 algebraic equation: one-step equation: Think 10.4 For one-step addition algebraic equations, use the inverse operation of subtraction to solve. (Hint: Addition and subtraction are inverse operations.) x (Subtract 7 from both sides of the equation.) x 3 Check: 3+7=10 (Substitute 3 for x in the original equation.) For one-step subtraction algebraic equations, use the inverse operation of addition to solve. m (Add 5 to both sides of the equation.) m 20 Check 5 (Substitute 20 for m in original equation.) Try It 10.4 Solve each addition and subtraction equation. Check answers. 1) x + 8 = 15 2) y + 20 = 10 3) m + 4 = 8 4) n ½ = 8 5) a 5.3 = 15 6) b 10 = 6 8

9 10.4 PRACTICE EXERCISES Solve the addition equations and check answers. 1) y + 8 = 16 2) m + 10 = 25 3) x + 6 = 18 4) a + 4 = 12 5) y 3 = 10 6) m 5 = 20 7) x ) b 10 = 12 9) Aleah bought some shoes on sale for $21, which were $7.00 off the regular price, write an equation to find the original price (p) of the shoes. Write and algebraic equation and solve. 10) Sondra shared 15 cookies from a bag of cookies. She had 32 cookies left in the bag. Write an algebraic equation to find the original number of cookies(c) she had in the bag. 9

10 10.5 ONE STEP MULTIPLICATION EQUATIONS Think 10.5 For multiplication equations, use the inverse operation of division to solve. 3x 3 15 (divide both sides of equation by 3.) 3 x 5 Check: y (Divide both sides by -9) y = 2 Check: -9 2=18 3x ( x 9 2 Check: 1 3 divide both sides of equation by 3.) y 15 (Divide both sides by 2.5) y = Check : Try It 10.5 Solve each multiplication equation. 1) 4y = 24 2) 8x = 32 3) 6y = 36 4) 2.5a = 10 5) 1 3b 6) 7a =

11 10.5 PRACTICE EXERCISES Solve each multiplication equation. Check answers. 1) 3y = 24 2) 5x = 25 3) 14m = 28 4) 12a = 144 5) 7m = 63 6) 15b = 45 7) 1 4x 2 8) 3.2y = 4 9) When you multiply a number (n) by 12, the product is 48. Write an equation to solve and find the number (n). 10) Lavry paid 5% tax on a purchase (p). The tax was $1.80. Write an equation to solve and find the total amount of the purchase. (Hint: 5% = 0.05). 11

12 10.6 ONE STEP DIVISION EQUATIONS Think 10.6 For division equations, use the inverse operation of multiplication to solve. x 10 5 x 50 (Multiply both sides of equation by 5.) 105 Check : x 50 x 1 2 x 3 (Multiply both sides of equation by 2.) x Check : x 6 x (Multiply both sides of equation by 6.) (2.5)(6) Check : x 15 Try It 10.6 Solve the one step division equations. Check answers. t n 1) 15 t = 2) 4 n = 12 7 m 81 3) 4.5 m = 4) 9 n = 4 n t 5) 36 t = 6) 56 7 x = 18 m 12

13 10.6 PRACTICE EXERCISES Solve the one step division equations. Check answers. 1) x 12 4 x = 2) y 3 11 y = 3) m m = 4) n n = 5) p $4 $9 p = 6) 7) y y = a 1 ( or.5) a = 5 2 8) b 8 7 b = 9) One out of 4 students in the day care class has the flu. 9 children are sick with the flu. Write an equation to and solve to find out how many students are in the class and solve. 10) A car wash package sold for $19.50 for 3 washes. Write an equation and solve to find out how much each car wash cost and solve. 13

14 10.7 ONE STEP EQUATIONS WITH A FRACTION COEFFICIENTS Think 10.7 For equations with a fraction coefficient multiply both sides of the equation by the reciprocal of the fraction coefficient. 2 3 y Fraction coefficient 24 To find the reciprocal, invert the numerator and denominator For, the reciprocol is For, the reciprocol is x 15(Multiply both sides by ) x x 25 Check:(Substitute 25 for x) Try It 10.7 Solve each fraction coefficient equation & check answers. 1) x x = 2) 3 y 15 y = 6 3) m m = 4) 3 n 36 n = 12 5) a a = 6) 6 b 54 b = 9 14

15 10.7 PRACTICE EXERCISES Solve each fraction coefficient equation & check answers. 1) 7 m 56 m = 8 2) 8 y 64 y = 10 3) 7 m 63 m = 10 4) 3 a 21 a = 9 5) 6 b 18 b = 8 6) 7 x 42 x = 8 7) 9 z 54 z = 10 8) 9 p 81 p = 11 9) 5 x 55 x = 6 10) 12 y 144 y = 15 15

16 CHAPTER 10 ANSWER KEY ) 3, x, 2 2) 6, x & y, 2 3) 3 & 2, x & y, 2 4) 7x and x 5) x 2 and 3x 2 6) n and 6n 7) 2x 2y 8) 5x 3 + 2x 2 9) 2m 3 + m 2 + m 10) 2x 2 + 3x ) y + 5 2) 2n 3) 3x 2 4) 2n 15 5) 9n - 6 6) 7 added to the product of 3 and n 7) The quotient of m divided by 2 minus 6 8) n increased by ) $37.88 n or $37.88 n 10) $25c ) 41 2) 45 3) 18 4) 147 5) 0 6) 1 7) $40 8) 10 9) 24 10) ) y = 8 2) m = 15 3) x = 12 4) a = -16 5) p = 13 6) m = 25 7) x = 8) b = ) p $7 = $21; p = $28 10) c 15 = 32; c= 47 cookies 16

17 10.5 1) y = 8 2) x = 5 3) m = 2 4) a = 12 5) m = 9 6) b = 3 7) x = ⅛ 8) y = ) n = 4 10) p = $ ) 64 2) 80 3) 90 4) 63 5) 24 6) 48 7) 60 8) 99 9) 66 10) 180 1) x = 48 2) y = 33 3) m = 400 4) n = 121 5) p = $36 6) y = 200 7) a = 2.5 8) b = 56 9) s = $36 10) c = $