advertisement

Algebraic Expressions Education's purpose is to replace an empty mind with an open one. Malcolm Forbes Expressions Math expressions represent a convenient way to translate verbal expressions What is the area of a rectangle? Length times Width If the length is 3 meters and the width is 2 meters, what is the area? A=LxW A = 3 x 2 = 6 m2 A, L and W are the variables. It is any letter that represents an unknown number. Algebraic Expression An algebraic expression is a quantity that contains numbers and variables. x + y , 3a2 a , 3x + 2y z Terms A term is a number, a variable, or a product of numbers and variables. Components of an Algebraic Expression Constant Variable term: fancy name for a number term: terms with letters 3xy – 4z + 17 Variable expression with 3 terms: 3xy, -4z, 17 2 variable terms and 1 constant term Example: Variable Terms Consist The of two parts variable(letter) part The number part Example: 2xy has a coefficient of 2 -6j has a coefficient of –6 W has a coefficient of 1 In expressions, there are many different ways to write multiplication. 1) 2) 3) 4) 5) ab a•b a(b) or (a)b (a)(b) axb We are not going to use the multiplication symbol any more. Why? Division, on the other hand, is written as: 1) x 3 2) x ÷ 3 Example of evaluating an expression. Evaluate 3xy – 2x + 7y when x = 2 and y = 3 3(2)(3) – 2(2) + 7(3) 18 – 4 + 21 14 + 21 35 The value of the expression is 35. Word Phrases as Algebraic Expressions Addition (+) sum plus added to more than increased by total Subtraction (–) difference minus subtract less than decreased by less Multiplication (·) product times multiply of double/triple Division () quotient divide shared equally among ratio of Word Phrases as Algebraic Expressions Write as an algebraic expression. Use x to represent “a number.” 5 decreased by a number In words: Translate: 5 5 decreased by – a number x Word Phrases as Algebraic Expressions Write as an algebraic expression. Use x to represent “a number.” The quotient of a number and 12 The quotient of In words: a number Translate: x and 12 12 x or 12 Write an algebraic expression for the following m increased by 5 m+5 7 times the product of x and t. 7(xt) or 7xt Write an algebraic expression for the following 11 less than 4 times a number. 4n - 11 two more than 6 times a number. 6n + 2 Which of the following expressions represents: 7 times a number decreased by 13 1. 2. 3. 4. 7x + 13 7x - 13 13 - 7x 13 + 7x Which one of the following expressions represents: 28 less than three times a number 1. 2. 3. 4. 28 - 3x 3x - 28 28 + 3x 3x + 28 Which of the following verbal expressions represents: 2x + 9 1. 2. 3. 4. 9 increased by twice a number a number increased by nine twice a number decreased by 9 9 less than twice a number Which of the following verbal expressions represents: x2 + 2x 1. 2. 3. 4. the sum of a number squared and twice a number the sum of a number and twice the number twice a number less than the number squared the sum of a number and twice the number squared Which of the following expressions represents: four less than the cube of a number 1. 2. 3. 4. 4 – x3 4 – 3x 3x – 4 x3 – 4 Terms Like terms Terms with the same variable part Same means same letter(s) and power(s) 2x, -5x ¾x2, 7x2 31xy, 4xy Terms We simplify variable expressions by combining like terms. To combine like terms, work with the coefficients of the like terms Terms We simplify variable expressions by combining like terms. To combine like terms, work with the coefficients of the like terms Combining Like Terms t + t + t+ t + t There are five variables which are like terms therefore we simply add them like we would if they were numbers. t + t + t+ t + t = 5t Combining Like Terms 4t + 3t + t = (4+3+1)t = 8t 2t2 + 8 – 5t2 Rearrange the variables so that all like terms are side by side. 2t2 – 5t2 + 8 (notice that the sign in front of the number came with the number) 2t2 – 5t2 + 8 = -3t2 + 8 Combining Like Terms 2t + 3t2 – 2t –t2 3t2 – t2 + 2t – 2t (collecting like terms) 2t2 + 0 2t2 Which figure below models the simplification of - 4x - 5 + 7x + 7 using these tiles? 1. 2. 3. 4. Combining Like Terms 3x + 5 – 9x = – 6x + 5 -5 +3b – 7 – 5b = – 2b - 12 3b – 5b = -2b -5 – 7 = -12 Simplifying Algebraic Expressions Rewrite using as few symbols as possible Use the distributive property if necessary to remove parentheses. Combine like terms More often than not will have numbers and letters in the final answer. Distributive Property Distributive Property a(b c) a b a c or a(b c) a b a c Objective - To use the distributive property to simplify numerical and variable expressions. Order of Operations 3(4 5) 3(9) 27 Distributive 3(4) 3(5) Property It works! 12 15 27 Why use the distributive property? 3(x 2) 3(x) 3(2) 3x 6 Simplify using the distributive property. 1) 5(x 3) 4) 4(3 y) 5 x 53 43 4 y 5x 15 12 4y 2) 6(y 7) 6 y 6 7 6y 42 5) 10(x 7) 10 x 10 7 10x 70 3) 3(m 8) 3 m 38 3m 24 6) 4(k 2) 4k 42 4k 8 Geometric Model for Distributive Property A wl 4 5 2 Two ways to find the area of the rectangle. As a whole As two parts 4 5 2 Geometric Model for Distributive Property 4 45 4 2 A wl 5 2 Two ways to find the area of the rectangle. As a whole As two parts same 45 4 2 4 5 2 4 5 2 4 5 4 2 Scientific Notation A short-hand way of writing large numbers without writing all of the zeros. The Distance From the Sun to the Earth 93,000,000 Step 1 Move decimal left Leave only one number in front of decimal Step 2 Write number without zeros Step 3 Count how many places you moved decimal If moved to left, make it positive If moved to right, make it negative Make that your power of ten 93,000,000 --Standard Form x 107 --Scientific Notation 9.3 Practice Problem Write in scientific notation. Decide the power of ten. 1) 2) 3) 4) 5) 10? 98,500,000 = 9.85 x 64,100,000,000 = 6.41 x 10? 279,000,000 = 2.79 x 10? 4,200,000 = 4.2 x 10? 0.000013 = 1.3 x 10? 9.85 x 107 6.41 x 1010 2.79 x 108 4.2 x 106 1.3 x 10-5 Complete Practice Problems Write in scientific notation. 1) 2) 3) 4) 50,000 7,200,000 802,000,000,000 0.000000000631 1) 5 x 104 2) 7.2 x 106 3) 8.02 x 1011 4) 6.31 x 10-10 Write in Standard Form Positive exponent → move to right Negative exponent → move to left x 106 9.01 x 104 3.95 x 10-3 6.27 6,270,000 90,100 0.00395