Section 1.8 Simplifying Expressions
But, first Commutative property: a + b = b + a; a * b = b * a Associative property: (a + b) + c = a + (b + c) (a * b) * c = a * (b * c) Distributive property: a * (b + c) = (a * b) + (a * c) Also, a * (b c) = (a * b) (a * c)
Simplifying expressions Identifying terms, like terms, and unlike terms Combining like terms Simplifying expressions containing parenthesis Write word phrases as algebraic expressions Vocabulary: term, numerical coefficient, combining like terms
Identifying Terms, Like Terms, and Unlike Terms A term is a number or a product of a number and variables possibly raised to powers In other words, a term is pretty much anything -y is a term: it is the product of -1 and y 2x 3 is a term: it is the product of 2 and x 3-5, 3xz 2 and 2/y are all terms. 0.8t is a term. The numerical coefficient of a term is the numerical factor. The numerical coefficient of -5 is -5 The numerical coefficient of 3xy 2 is 3 (not 2, it is a exponent, and exponents are not factors)
Identify the numerical coefficient of each term 9x -3y -x 2.7x 2 y x 5
Like terms and unlike terms Terms with the same variables raised to the exact same powers are called like terms. 5x and 9x are like terms. The variable for both is x 5x and 9x 2 are unlike terms. The variable for 5x is x. The variable for 9x 2 is x 2 : different powers 6x and 7y are unlike terms: different variables. 5x 2 y and -4x 2 y are like terms: same variables and powers -3ab 2 and 5a 2 b are unlike terms: in the first one, b is squared, and in the second, a is squared
How about? Are 9x 2 y and -6yx 2 like terms? Based on the commutative property, -6yx 2 can be written as -6x 2 y Therefore, 9x 2 y and -6yx 2 are like terms Each variable and its exponent must match exactly, but the order they are written does not matter
Indicate whether the terms are like terms 6x, -3x -xy 2, -x 2 y 5ab, (-1/2)ba 2x 3 yz 2, -x 3 yz 3 2zx 3 yz 2, -x 3 yz 3
Combining like terms Why the big deal about like terms? In a complicated algebraic expression, like terms can be combined together to make things simpler The Distributive property makes this possible For example, an expression has like terms 6x and 2x 6x + 2x = (6 + 2)x = 8x Also: -y 2 + 5y 2 = (-1 + 5)y 2 = 4y 2 Simplifying the sum or difference of like terms is called combining like terms.
Simplify each expression by combining any like terms 7z 2z + 4-9y + 2-1 + 6 + y 7 1.6x 5 + 0.9x 2 0.3x 5 8x 2 y 4yx 2 + 2xy 2xyx
Simplifying expressions containing like terms Isn t simplifying just a matter of removing parenthesis? (3a + 2) = +1(3a + 2) = +1(3a) + (+1)(2) = 3a + 2 Big deal? Ok, but throw a negative sign in there: -(3a + 2) = -1(3a + 2) = -1(3a) + (-1)(2) = -3a 2 In order to remove parenthesis, you have to apply the Distributed property -(9x + y 2z + 6) = -1 * (9x + y 2z + 6) Distribute: -1(9x) + (-1)(y) + (-1)(-2z) + (-1)(6) = -9x + -y + 2z + -6 = -9x y + 2x - 6
Helpful hint: page 71 If a negative - sign precedes parenthesis, the sign of each term within the parenthesis will change when the parenthesis is removed -(2x + 1) = -2x 1 -(x 2y) = -x + 2y -(-5x + y 0.3z) = 5x y + 0.3z
Use the distributive property to remove the parenthesis 3(x + 6) -(-5m + 6n 2p) 1/3(6x 9) 14(2x + 6) 4 10a 5 2(a 3)
Writing algebraic expressions Twice a number, plus 6 (2 * x) + 6 = 2x + 6 The difference of a number and 4, divided by 7 (x 4) 7 Five plus the sum of a number and 1 5 + (x + 1) = 5 + x + 1 = 6 + x Four times the sum of a number and 3 4 * (x + 3) = (4 * x) + (4 * 3) = 4x + 12
Write each phrase as an algebraic expression. Add -4y + 3 to 6y + 9 Subtract 2x -1 from 3x + 7 Triple a number, decreased by 6 Six times the sum of a number and two
Chapter 1 Review
Vocabulary Check A mathematical statement that two expressions are equal is called The of a number is the distance between it and 0 on the number line The number in a fraction above the fraction bar is called The number on the bottom of the fraction is the In 2 3, the 2 is called the and the 3 is the The is the product of a number and variables raised to powers The of a term is its numerical factor Terms with the same variables raised to the same powers are called
Insert <, >, or = in the spaces 8 10-4 - 5 12/2-8 -7-8 -9-9 - -1-1 -14 -(-14) 1.2 1.02-3/2-3/4
Translate each statement into symbols Four is greater than or equal to negative three Six is not equal to five 0.03 is less than 0.3
{-6, 0, 1, 1½, 3, π, 9.62, 2} Which are integers? Which are rational numbers? Which are irrational numbers? Which are real numbers?
Simplify (solve) each expression 6 * 3 2 + 2 * 8 68 5 * 2 3 3(1 + 2 * 5) + 4 8 + 3(2 * 6 1) 5[3(2 + 5) 5]
Translate each word statement to symbols The difference of twenty and twelve is equal to the product of two and four The quotient of nine and two is greater than negative five
Evaluate each when x=6, y=2, & z=8 2x + 3y x(y + 2z) (x/y) + (z/2y) (x 2 ) 3y 2
Decide whether the given number is a solution to the given equation 7x 3 = 18; 3 3x 2 + 4 = x 1; 1
Find the additive inverse or opposite to each number -9 2/3-2 - -7
Solve each problem: -15 + 4 = -6 + (-11) = 1/16 + (-1/4) = -8 + -3 = -4.6 + (-9.3) = 6 20 = -3.1 8.4 = -6 (-11) = -21 16 + 3(8 2) = 2 + 3(4) =
Find each multiplicative inverse or reciprocal -6 3/5 1¼ 0.25 0/12
Simplify (solve) each expression 6(-8) (-2)(-14) -18/-6 42/-3-3(-6)(-2) (-4)(-3)0(-6) (3(-2) 2 5)/-14
Name the property illustrated -6 + 5 = 5 + (-6) 6 * 1 = 6 3(8 5) = 3 * 8 + 3 * (-5) 4 + (-4) = 0 2 + (3 + 9) = (2 + 3) + 9 2 * 8 = 8 * 2 (3 * 8) * 4 = 3 * (8 * 4) 4 * ¼ = 1 4(8 + 3) = 4(3 + 8) 5(2 + 1) = 5*2 + 5*1
Simplify each expression 5x x + 2x 0.2z 4.6z 7.4z 1/2x + 3 + 7/2x 5 (4/5)y + 1 + (6/5)y + 2 2(n 4) + n - 10