WARM UP Simplify 1. –5 – 7 2. 7 + (– 4) −12 3. –6  7 3 (−5)(−4) 1. –5 – 7 2. 7 + (– 4) 3. –6  7 (−5)(−4) −3/7  ⅘ −⅘ ÷ 11/3 −12 3 −42 20 −12/35 −12/55

ALGEBRAIC EXPRESSIONS & PROPERTIES OF NUMBERS

OBJECTIVES Evaluate algebraic expressions Write equivalent expressions Solve problems algebraically

KEY TERMS & CONCEPTS Additive identity Evaluating Algebraic expressions Associative Properties Commutative Properties Constant Equivalent expressions Evaluating Multiplicative identity Substitute a number Value variables

ALGEBRAIC EXPRESSIONS In algebra we use letters to represent numbers. For example, in the formula for the area of a circle: A stands for the area and r stands for the radius. A and r can represent various numbers, so they are called variables The symbol can represent only one number (3.14159. . . .), so it is called constant.

ALGEBRAIC EXPRESSIONS Algebraic expressions consist of numerals, variables and other mathematical symbols such as + or √. Some examples of algebraic expressions: x + 5 |y – 8| When we substitute numbers for the variables in an expression and then calculate the number, we get the value of the expression for those replacements We say that we are evaluating the expression.

EXPRESSING ALGEBRAIC EXPRESSIONS Meaning Operation 5(n) 5  n 5n 5 times n Multiplication 14 ÷ y 14 divided by y Division 6 + c 6 plus c Addition 8 – x 8 minus x Subtraction

EXAMPLES a. 12  n = 13  3 substitute 3 for n Evaluate the expression when n = 3 a. 12  n = 13  3 substitute 3 for n = 39 multiply b. = = 3 substitute 3 for n divide c . n – 1 = 3 – 1 = 2 substitute 3 for n subtract d. n + 8 = 3 + 8 = 11 substitute 3 for n add

EXAMPLES 1. Evaluate 2y + x for x = 3 and y = 5 2y + x = 2  5 + 3 Substituting 3 for x and 5 for y = 10 + 3 Calculating = 13 The value of the expression is 13. 2. Evaluate –(-x) for x = -7 -(-x) = -(-(-7)) Substituting -7 for x = -(7) Calculating within the parenthesis PEMDAS = -7 The value of the expression is -7.

MORE EXAMPLES 1. Evaluate |x| + 2|y| for x = 15 and y = -10 |x| + 2|y| = |15|+ 2|-10| Substituting 15 for x and -10 for y = 15 + 2  10 Finding absolute value = 15 + 20 = 35 The value of the expression is 35.

TRY THIS… Evaluate each expression 5x – y for x = 10 and y = 5 -(-y) for y = -8 |x| - 2|y| for x = -16 and y = -4

APPLY ORDER OF OPERATIONS Mathematicians have created an order of operations to evaluate an expression involving more than one operation PEMDAS Parenthesis Exponents Multiply/divide Add/subtract

ORDER OF OPERATIONS STEPS Step 1 – Evaluate expressions inside grouping symbols: Parenthesis Step 2 – Evaluate powers: Exponents Step 3 – Multiply and Divide from left to right Step 4 – Add and Subtract from left to right PEMDAS Parenthesis – exponents – multiplication/division – addition/subtraction “Please excuse my dear aunt sally”

EXAMPLES Evaluate the expression 27 ÷ 3 2 - 3 Step 1 There are no grouping symbols (parenthesis), so go to step 2 Step 2 Evaluate powers – 27 ÷ 3  2 – 3 = 27 ÷ 9  2 - 3 (Exponents) Step 3 Multiply and divide from left to right 27 ÷ 9  2 – 3 = 3  2 - 3 Step 4 Add and subtract from left to right 3  2 - 3 = 6 – 3 = 3 The value of this expression is 3

TRY THIS… Evaluate using PEMDAS when x = 9 and y = -4 4

EQUIVALENT EXPRESSIONS The subtraction theorem tells us that the expressions m – n and m + (-m) will always have the same value whenever we make the same substitutions in both expressions. Equivalent expressions always have the same value for all acceptable replacements (substitutions).

EXAMPLES 1. Use the subtraction theorem to write equivalent expressions. 4y – x = 4y + (-x) Adding an inverse 3p + 5q = 3p – (-5q) Using the subtraction theorem in reverse.

TRY THIS… Use the subtraction theorem to write equivalent expressions -5x – 3y 17m – 45 -6p + 5t -5x + (-3y) 17m + (-45) – 6p – (-5t)

NUMBER PROPERTIES Number properties help us identify equivalent expressions. Commutative Properties tells us that we can change order when adding or multiplying and obtain an expression equivalent to the original one. Example: a + b = b + a 1 + 3 = 3 + 1 a x b = b x a 3 x 1 = 1 x 3 Associative Property tells us that we can change grouping when adding or multiplying and obtain an equivalent expression. Example: a + (b + c) = (a + b) + c 5 + (3 + 2) = (5 + 3) + 2 a × (b × c) = (a × b) × c 5 × (3 × 2) = (5 × 3) × 2

EXAMPLES 1. Use the commutative property of addition to write an expression equivalent to 3x + 4y. 3x + 4y = 4y + 3x Changing order 2. Use the associative property of multiplication to write an expression equivalent to 3x(7y  9z) 3x(7y  9z) = (3x  7y)  9z Changing grouping

MORE EXAMPLES We use both the commutative & associative properties to write equivalent expressions. 3. Use the commutative and associative properties of addition to write an expression equivalent to (5/x + 2y) + 3z 5/x + 2y + 3z = 5/x +(2y + 3z) Using the associative property = 5/x +(3z + 2y) Using the commutative property

TRY THIS…… Use the associative property of addition to write an expression equivalent to (8m + 5n) + 6p. Use the commutative property of multiplication to write an expression equivalent to (17x)(-9t). Use the commutative and associate properties of multiplication to write an expression equivalent to 9p(4q  16r)

NUMBER PROPERTIES a –a = 0 Identity Property: The number “0” is the additive identity and the number “1” is the multiplicative identity. a –a = 0 Example: Write an expression equivalent to 4x – 2 by adding o. Use 7y – 7y for 0: 4x – 2 = 4x - 2 + 7y – 7y adding 0 = 4x + 7y – 2 – 7 y rearranging The expression 4x – 2 and 4x + 7y – 2 – 7y are equivalent

EXAMPLES 1. Write an expression equivalent to x/3y. Use 8/8 for 1 Multiplying by 1 Multiplying numerators and denominators. The expression x/3y and y are equivalent. They will represent the same number for any acceptable replacements for x and y.

TRY THIS…… Write an expression equivalent to 8a – b by adding 0. Use x – x for 0. Write an expression equivalent to 19t/3x. Use 9/9 for 1. 8a + x – b – x

CH. 1.3 HOMEWORK Textbook pg. 18 #4, 12, 16, 24, 30, 34, 36 & 43