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Properties of Addition & Multiplication

The document discusses various properties of real numbers including the commutative, associative, identity, inverse, zero, and distributive properties. It also covers topics such as combining like terms, translating word phrases to algebraic expressions, and simplifying algebraic expressions. Examples are provided to illustrate each concept along with explanations of key terms like coefficients, variables, and like terms.

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Commutative Property of Addition /Multiplication  The order in which numbers are added does not change the sum. 5+3=3+5  For any numbers a and b a+b=b+a  The order in which numbers are multiplied does not change the product 2·4=4·2  For any numbers a and b a·b=b·a Commutative – switching places or interchanging  Think of the commutative property as physically changing places, they commute or substitute one for the other.
Associative Properties of Addition/Multiplication  The way in which addends are grouped does not change the sum. (2 + 4) + 6 = 2 + (4 + 6) For any numbers a, b, and c. (a + b) + c = a + (b + c)  The way in which numbers are grouped does not change the product. (6 · 3) · 7 = 6 · (3 · 7) For an numbers a, b, and c, (a · b) · c = a · (b · c)  The associative property can be thought of as “friendships” (associations). The parentheses show the grouping of two friends. They don’t physically move, they simply change the one with whom they are associating.
Identity Properties of Addition/Multiplication  The sum of a number and zero is the number. 6+0=6 For any number a, a+0=a  The product of a number and one is the number. 6·1=6 For any number a, a·1=a  The identity element here stays the same, so if “I” add zero “I” remain the same. If “I” multiply by one, “I” remain the same.
Multiplicative Property of Zero  The product of a number and zero is zero. 5·0=0 For any number a, a·0=0  The sum of a number and its opposite are equal to zero. 5 + (-5) = 0 For any number a, a + (-a) = 0  The product of a number and its multiplicative inverse equals one. 2·½=1 For any number a, a · 1/a = 1  Think of the inverse property as what would you need to add (multiply) to this number to turn it into an identity element? The additive inverse is the negative of the number, and the multiplicative inverse is one divided by the number.
Distributive Property  The sum of 2 addends (b + c) multiplied by a number (a) is the sum of the product of each addend and the number. 3(4 + 5) = 3(4) + 3(5) For any number a, b, and c, a(b + c) = ab + ac or (b + c)a = ab + bc The expression a(b + c) is read “a times the quantity b plus c” or “a times the sum of b and c”  Using the distributive property lets you multiply each element inside the parentheses by the element outside the parentheses. Consider the problem to the left. The number in front of the parentheses is “looking” to distribute (multiply) its value with all of the terms inside the parentheses.
Properties of Real Numbers Property Example 1 Commutative Property of Addition a+b=b+a 2+3=3+2 2 Commutative Property of Multiplication a·b=b·a 2 · (3) = 3 · (2) 3 Associative Property of Addition a + (b + c) = (a + b) + c 2 + (3 + 4) = 2 + (3 + 4) 4 Associative Property of Multiplication a · (b · c) = (a · b) · c 2 · (3 · 4) = (2 · 3) · 4 5 Distributive Property a · (b · c) = a · b + a · c 2 · (3 + 4) = 2 · 3 + 2 · 4 6 Identity Property of Addition a+0=a 3+0=3 7 Identity Property of Multiplication a·1=a 3·1=3 8 Additive Inverse Property a + (-a) = 0 3 + (-3) = 0 9 Multiplicative Inverse Property a · (1/a) = 1 3 · (1/3) = 1 10 Property of Zero a·0=0 5·0=0
The Language of Algebra  Algebra, like any language, is a language of symbols. It is the language of math and must be learned as any other language. You know the symbols of division and addition, so you can write the blood-pressure relationship as: age ÷ 2 + 110 In arithmetic, you could write: □ ÷ 2 + 110  In algebra, we use variables, letters that represent unknown values. In this case the letter x: X ÷ 2 + 110 This is known as a algebraic expression.  If Samantha is 18 years old, she could estimate her blood pressure by evaluating the expression, 18 ÷ 2 + 110 a ÷ 2 + 110 = (18) ÷ 2 + 110 substitute 18 for a = 9 + 110 order of operations, division first = 119
When reading a verbal sentence and writing an algebraic expression to represent it, there are words and phrases that suggest the operations to use. Addition Plus Sum More than Increased by Total In all Subtraction Minus Difference Less than Subtract Decreased by Multiplication Times Product Multiplied Each Of Division Divided quotent Translating Word Phrases into Math Expressions  While the table on the previous slide gives you an idea about phrases that translate to math operations, being able to identify the key words that determine the operations (+, -, ·, ÷) that will be used to solve problems takes practice.
Write an expression for each phrase. 1) 2) 3) 4) 5) 6) 7) 8) A number n divided by 5 The sum of 4 and a number y 3 times the sum of a number b and 5 The product of a number n and 9 The sum of 11 times a number s and 3 7 minus the product of 2 and a number x 6 less than a number x 7 times the sum of x and 6 Write an algebraic expression to evaluate the word problem: 1) 2) Samantha purchased a 200-minute calling card and called her father from college. After talking with him for t minutes, how many minutes did she have left on her card? Write and solve an expression to represent the number of minutes remaining on the calling card. Jared worked for h hours at $5 per hour. Write an expression to determine how much money Jared earned. How much money will Jared earn if he works a total of 18 hours?
Combining Like Terms  Term – The parts of an expression that are added or subtracted. (x + 2) (2x – 4)  Like terms – 2 or more terms that have the same variable raised to the same power. (in the expression 3a + 5b + 12a, 3a and 12a are like terms.)  To simplify an expression – Perform all possible operations, including combining like terms. Add or Multiply? x + x x x x + y x 1x + 1x = 2x x x 1x + 1y = x + y y x y
 A procedure frequently used in algebra is the process of combining like terms. This is a way to “clean-up” an equation and make it easier to solve.  For example, in the algebraic expression 4x + 3 + 7y, there are three terms: 4x, 3, and 7y. Remember the 4 and 7 are coefficients.  Let’s say we are given the equation below. It looks very complicated, but if we look carefully, everything is either a constant (number), or the variable x with a coefficient (4x). Remember, a coefficient is the number by which a variable is being multiplied (the 4 in 4x is the coefficient)
 The “like terms” in the equation are ones that have the same variable. All constants are like terms as well.  This means 15, 10, 6, and -2 are all like terms, and the other is 4x, -3x, 5x, and 3x. To combine them is pretty easy, you just add them together and make sure they are all on the same side of the equation.  Since the 15 and 10 are both constants we combine them to get 25. The 4x and -3x each have the same variable (x), so we can add them to get 1x. Doing the same on the other side we arrive at 25 + 1x = 4 + 8x. The process is still not finished.  There are still some like terms, but they are on opposite sides of the equal sign. Since we can do the same thing to both sides we just subtract 4 from each side and subtract 1x from each side. What remains is 21 = 7x.
 Now it’s just a simple process of dividing by seven on each side and we arrive at our answer of x = 3. Combining like terms enables you to take that huge mess of an equation and make it something much more obvious to solve. Simplify Algebraic Expressions by combining like terms. Simplify: 6(n + 5) – 2n = 6 (n) + 6(5) – 2n = Distributive Property 6n + 30 – 2n = 6n and 2n are like terms 4n + 30 Combine coefficients 6 – 2 = 4 Remember that a term like “x” has a coefficient of 1, so terms such as x, n, or y can be written as 1x, 1n, or 1y.
Example 1: 2a + 5b + 5 – a + 3 How many terms are in this expression? What are the like terms? Simplify by combining like terms. a + 5b + 8 Example 2: 2 + 8(3y + 5) – y What would be the first step in simplifying the expression? Use the Distributive Property to simplify 8(3y + 5), 8(3y) + 8(5), 24y + 40 Combine like terms. 2 + 24y + 40 –y 42 + 23y
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