Chapter 5.5 Slides 1-19 Math 7

Algebraic Expressions An algebraic expression is a collection of real numbers, variables, grouping symbols and operation symbols. Here are some examples of algebraic expressions.

Consider the example: The terms of the expression are separated by addition. There are 3 terms in this example and they are . The coefficient of a variable term is the real number factor. The first term has coefficient of 5. The second term has an unwritten coefficient of 1. The last term , -7, is called a constant since there is no variable in the term.

Let’s begin with a review of two important skills for simplifying expression, using the Distributive Property and combining like terms. Then we will use both skills in the same simplifying problem.

Distributive Property To simplify some expressions we may need to use the Distributive Property Do you remember it? Distributive Property a ( b + c ) = ba + ca

Examples Example 1: 6(x + 2) Distribute the 6. 6 (x + 2) = x(6) + 2(6) = 6x + 12 Example 2: -4(x – 3) Distribute the –4. -4 (x – 3) = x(-4) –3(-4) = -4x + 12

Practice Problem Try the Distributive Property on -7 ( x – 2 ) . Be sure to multiply each term by a –7. -7 ( x – 2 ) = x(-7) – 2(-7) = -7x + 14 Notice when a negative is distributed all the signs of the terms in the ( )’s change.

Examples with 1 and –1. Example 3: (x – 2) = 1( x – 2 ) = x(1) – 2(1) Notice multiplying by a 1 does nothing to the expression in the ( )’s. Example 4: -(4x – 3) = -1(4x – 3) = 4x(-1) – 3(-1) = -4x + 3 Notice that multiplying by a –1 changes the signs of each term in the ( )’s.

Like Terms Like terms are terms with the same variables raised to the same power. Hint: The idea is that the variable part of the terms must be identical for them to be like terms.

Examples Like Terms 5x , -14x -6.7xy , 02xy The variable factors are identical. Unlike Terms 5x , 8y The variable factors are not identical.

Combining Like Terms Recall the Distributive Property a (b + c) = b(a) +c(a) To see how like terms are combined use the Distributive Property in reverse. 5x + 7x = x (5 + 7) = x (12) = 12x

Example All that work is not necessary every time. Simply identify the like terms and add their coefficients. 4x + 7y – x + 5y = 4x – x + 7y +5y = 3x + 12y

Collecting Like Terms Example

Both Skills This example requires both the Distributive Property and combining like terms. 5(x – 2) –3(2x – 7) Distribute the 5 and the –3. x(5) - 2(5) + 2x(-3) - 7(-3) 5x – 10 – 6x + 21 Combine like terms. - x+11

Simplifying Example

Simplifying Example Distribute.

Simplifying Example Distribute.

Simplifying Example Distribute. Combine like terms.

Simplifying Example Distribute. Combine like terms.

Evaluating Expressions Evaluate the expression 2x – 3xy +4y when x = 3 and y = -5. To find the numerical value of the expression, simply replace the variables in the expression with the appropriate number. Remember to use correct order of operations.

Example Evaluate 2x–3xy +4y when x = 3 and y = -5. Substitute in the numbers. 2(3) – 3(3)(-5) + 4(-5) Use correct order of operations. 6 + 45 – 20 51 – 20 31

Evaluating Example

Evaluating Example Substitute in the numbers.

Evaluating Example Substitute in the numbers.

Evaluating Example Substitute in the numbers. Remember correct order of operations.

Common Mistakes Incorrect Correct

Your Turn Find the product (-8)(3) (20)(-65) (-15) Simplify the variable expression (-3)(-y) 5(-a)(-a)(-a)

Your Turn Evaluate the expression: -8x when x = 6 3x2 when x = -2 -4(|y – 12|) when y = 5 -2x2 + 3x – 7 when x = 4 9r3 – (- 2r) when r = 2

Your Turn Solutions -24 -1300 -9 3y -5a3 -48 12 -28 -27 76

Find the product. a. (9)(–3) b. c. (–3)3 d. -27 (–4)(–6) 24 (–3)(–3)(–3) 1(–3)(–5) (9)(–3) (–3)(–5) –27 15

Find the product. a. (–n)(–n) b. (–4)(–x)(–x)(x) c. –(b)3 d. (–y)4 Two negative signs: n2 Three negative signs: –4x3 One negative sign: –(b)(b)(b) = –b3 Four negative signs: (–y)(–y)(–y)(–y) = y4 SUMMARY: An even number of negative signs results in a positive product, and an odd number of negative signs results in a negative product.

Extra Example 3 Evaluate the expression when x = –7. a. 2(–x)(–x) OR simplify first: 2(-7)2 2(49) 98

Extra Example 3 (cont.) Evaluate the expression when x = –7. b. OR use the associative property:

Checkpoint Find the product. 1. (–2)(4.5)(–10) 2. (–4)(–x)2 3. Evaluate the expression when x = –3: (–1• x)(x) 90 –4x2 –9

Properties of Real Numbers Commutative Associative Distributive Identity + × Inverse + ×

Commutative Properties Changing the order of the numbers in addition or multiplication will not change the result. Commutative Property of Addition states: 2 + 3 = 3 + 2 or a + b = b + a. Commutative Property of Multiplication states: 4 • 5 = 5 • 4 or ab = ba.

Associative Properties Changing the grouping of the numbers in addition or multiplication will not change the result. Associative Property of Addition states: 3 + (4 + 5)= (3 + 4)+ 5 or a + (b + c)= (a + b)+ c Associative Property of Multiplication states: (2 • 3) • 4 = 2 • (3 • 4) or (ab)c = a(bc)

Distributive Property Multiplication distributes over addition.

Additive Identity Property There exists a unique number 0 such that zero preserves identities under addition. a + 0 = a and 0 + a = a In other words adding zero to a number does not change its value.

Multiplicative Identity Property There exists a unique number 1 such that the number 1 preserves identities under multiplication. a ∙ 1 = a and 1 ∙ a = a In other words multiplying a number by 1 does not change the value of the number.

Additive Inverse Property For each real number a there exists a unique real number –a such that their sum is zero. a + (-a) = 0 In other words opposites add to zero.

Multiplicative Inverse Property For each real number a there exists a unique real number such that their product is 1.

Let’s play “Name that property!”

State the property or properties that justify the following. 3 + 2 = 2 + 3 Commutative Property

State the property or properties that justify the following. 10(1/10) = 1 Multiplicative Inverse Property

State the property or properties that justify the following. 3(x – 10) = 3x – 30 Distributive Property

State the property or properties that justify the following. 3 + (4 + 5) = (3 + 4) + 5 Associative Property

State the property or properties that justify the following. (5 + 2) + 9 = (2 + 5) + 9 Commutative Property

Commutative Property of Addition 2. Which Property? 3 + 7 = 7 + 3 Commutative Property of Addition

Identity Property of Addition 3. Which Property? 8 + 0 = 8 Identity Property of Addition

Commutative Property of Multiplication 5. Which Property? 6 • 4 = 4 • 6 Commutative Property of Multiplication

Identity Property of Multiplication 11. Which Property? 5 • 1 = 5 Identity Property of Multiplication

Identity Property of Addition 25. Which Property? 51/7 + 0 = 51/7 Identity Property of Addition

Inverse Property of Addition 40. Which Property? a + (-a) = 0 Inverse Property of Addition