# Algebraic Expressions. Education's purpose is to replace an empty mind with an open one. Malcolm Forbes.

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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 = L x W A = 3 x 2 = 6 m 2 A, L and W are the variables. It is any letter that represents an unknown number.

An algebraic expression is a quantity that contains numbers and variables. x + y, 3a 2  a, 3x + 2y  z A term is a number, a variable, or a product of numbers and variables. Terms Algebraic Expression

Components of an Algebraic Expression  Constant term: fancy name for a number  Variable term: terms with letters  Example: 3xy – 4z + 17  Variable expression with 3 terms:  3xy, -4z, 17  2 variable terms and 1 constant term

Variable Terms  Consist of two parts  The 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)ab 2)a b 3)a(b) or (a)b 4)(a)(b) 5)a x b We are not going to use the multiplication symbol any more. Why?

Division, on the other hand, is written as: 1) 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 (+) Subtraction (–) Multiplication (·) Division (  ) sum plus added to more than increased by total difference minus subtract less than decreased by less product times multiply of double/triple quotient divide shared equally among ratio of

Word Phrases as Algebraic Expressions Write as an algebraic expression. Use x to represent “a number.” In words: 5 decreased by a number 5 Translate: 5 decreased bya 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 In words:a number Translate: x and12  The quotient of

Write an algebraic expression for the following m increased by 5 7 times the product of x and t. m + 5 7(xt) or 7xt

Write an algebraic expression for the following 11 less than 4 times a number. two more than 6 times a number. 4n - 11 6n + 2

Which of the following expressions represents: 7 times a number decreased by 13 1. 7x + 13 2. 7x - 13 3. 13 - 7x 4. 13 + 7x

Which one of the following expressions represents: 28 less than three times a number 1. 28 - 3x 2. 3x - 28 3. 28 + 3x 4. 3x + 28

Which of the following verbal expressions represents: 2x + 9 1. 9 increased by twice a number 2. a number increased by nine 3. twice a number decreased by 9 4. 9 less than twice a number

Which of the following verbal expressions represents: x 2 + 2x 1. the sum of a number squared and twice a number 2. the sum of a number and twice the number 3. twice a number less than the number squared 4. 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. 4 – x 3 2. 4 – 3x 3. 3x – 4 4. x 3 – 4

Terms  Like terms  Terms with the same variable part  Same means same letter(s) and power(s) 2x, -5x ¾x 2, 7x 2 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 2t 2 + 8 – 5t 2 Rearrange the variables so that all like terms are side by side. 2t 2 – 5t 2 + 8 (notice that the sign in front of the number came with the number) 2t 2 – 5t 2 + 8 = -3t 2 + 8

Combining Like Terms 2t + 3t 2 – 2t –t 2 3t 2 – t 2 + 2t – 2t (collecting like terms) 2t 2 + 0 2t 2

Which figure below models the simplification of - 4x - 5 + 7x + 7 using these tiles? 1. 2. 3. 4.

Combining Like Terms 3x + 5 – 9x = -5 +3b – 7 – 5b = – 6x + 5 -5 – 7 = -12 3b – 5b = -2b – 2b - 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 or Distributive Property

Objective - To use the distributive property to simplify numerical and variable expressions. Order of Operations Distributive Property It works! Why use the distributive property?

Simplify using the distributive property. 1) 2) 3) 4) 5) 6)

Geometric Model for Distributive Property Two ways to find the area of the rectangle. 4 52 As a wholeAs two parts

Geometric Model for Distributive Property Two ways to find the area of the rectangle. 4 52 As a wholeAs two parts same

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 9.3 x 10 7 --- Scientific Notation

Practice Problem 1) 98,500,000 = 9.85 x 10 ? 2) 64,100,000,000 = 6.41 x 10 ? 3) 279,000,000 = 2.79 x 10 ? 4) 4,200,000 = 4.2 x 10 ? 5) 0.000013 = 1.3 x 10 ? Write in scientific notation. Decide the power of ten. 9.85 x 10 7 6.41 x 10 10 2.79 x 10 8 4.2 x 10 6 1.3 x 10 -5

Complete Practice Problems 1) 50,000 2) 7,200,000 3) 802,000,000,000 4) 0.000000000631 Write in scientific notation. 1) 5 x 10 4 2) 7.2 x 10 6 3) 8.02 x 10 11 4) 6.31 x 10 -10

Write in Standard Form 6.27 x 10 6 9.01 x 10 4 3.95 x 10 -3 6,270,000 90,100 0.00395 Positive exponent→ move to right Negative exponent → move to left

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