Presentation on theme: "Literal Equations, Numbers, and Quanties"— Presentation transcript:
1 Literal Equations, Numbers, and Quanties Module 1 Lesson 2
2 What is a Literal Equation? A Literal Equation is an equation with two ormore variables.You can "rewrite" a literal equation to isolate any one of the variables using inverse operations.When you rewrite literal equations, you may have to divide by a variable or variable expression.
3 Solving a Literal Equation Step 1Locate the variable you are asked to solvefor in the equation.Step 2Identify the operations on this variable andthe order in which they are applied.Step 3Use inverse operations to undo operationsand isolate the variable.
4 Example: Solving Literal Equations A. Solve x + y = 15 for x.x + y = 15Since y is added to x, subtract yfrom both sides to undo theaddition.–y –yx = –y + 15B. Solve pq = x for q.pq = _x_p pSince q is multiplied by p, divideboth sides by p to undo themultiplication.
5 Your Turn: Solve 5 – b = 2t for t. 5 – b = 2t Locate t in the equation.Since t is multiplied by 2, divide both sides by 2 to undo the multiplication.
6 Your Turn! Solve for the indicated variable. 1. 2. for h 3. 2x + 7y = 14 for y4.for hP = R – C for Cfor m5.for C
7 Your Turn Solutions H = 3V A 2. y = 14 – 2x 7 3. C = R – P 4. m = x(k – 6 )5. C = Rt + S
8 Example: ApplicationThe formula C = d gives the circumference of a circle C in terms of diameter d. The circumference of a bowl is 18 inches. What is the bowl's diameter? Leave the symbol in your answer.Locate d in the equation.Since d is multiplied by , divide bothsides by to undo the multiplication.Now use this formula and the information given in the problem.
9 Example: ContinuedThe formula C = d gives the circumference of a circle C in terms of diameter d. The circumference of a bowl is 18 inches. What is the bowl's diameter? Leave the symbol in your answer.Now use this formula and the information given in the problem.The bowl's diameter is inches.
10 The Real Number SystemEvery Real Number is either rational or irrational.We refer to these sets as subsets of the real numbers, meaning that all elements in each subset are also elements in the set of real numbers.
12 Consider the following set of numbers. ExampleConsider the following set of numbers.List the numbers in the set that are:Natural NumbersWhole NumbersIntegersRational NumbersIrrational NumbersReal numbers
13 Consider the following set of numbers. ExampleConsider the following set of numbers.List the numbers in the set that are:Natural Numbers: √16 = 4, so that is the only Natural NumberWhole Numbers: 0 , √16Integers: -3, 0, √16Rational Numbers: -3, 0, ½ , .95, √16Irrational Numbers: √8Real numbers: All of the numbers listed above!
14 Properties of Real Numbers PropertyAdditionMultiplicationCommutativea + b = b + aab = baAssociative(a + b) + c = a + (b + c)(ab)c = a(bc)Identitya + 0 = aa * 1 = aInversea + (-a) = 0a * 1/a = 1Distributivea(b + c) = ab + ac
15 Communitive Property Rules: a+b = b+a ab = ba 1+2 = 2+1 (2x3) = (3x2) It doesn’t matter how you swap addition or multiplication around…the answer will be the same!Rules:Commutative Property of Additiona+b = b+aCommutative Property of Multiplicationab = baSamples:Commutative Property of Addition1+2 = 2+1Commutative Property of Multiplication(2x3) = (3x2)The next slide will discuss how these do not apply to subtraction and division.
16 Associative Property Rules: (a+b)+c = a+(b+c) (ab)c = a(bc) It doesn’t matter how you group (associate) addition or multiplication…the answer will be the same!Rules:Associative Property of Addition(a+b)+c = a+(b+c)Associative Property of Multiplication(ab)c = a(bc)Samples:Associative Property of Addition(1+2)+3 = 1+(2+3)Associative Property of Multiplication(2x3)4 = 2(3x4)Later we will discuss how these do not apply to subtraction and division.
17 Identity Property Samples: Rules: 3+0=3 a+0 = a a(1) = a 2(1)=2 What can you add to a number & get the same number back? ZEROWhat can you multiply a number by and get the number back? ONESamples:Identity Property of Addition3+0=3Identity Property of Multiplication2(1)=2Rules:Identity Property of Additiona+0 = aIdentity Property of Multiplicationa(1) = a
18 Inverse Property Rules: a+(-a) = 0 a(1/a) = 1 Samples: 3+(-3)=0 Think opposites!The Inverse property uses the inverse operation to get to the identity!Rules:Inverse Property of Additiona+(-a) = 0Inverse Property of Multiplicationa(1/a) = 1Samples:Inverse Property of Addition3+(-3)=0Inverse Property of Multiplication2(1/2)=1
19 Distributive Propety Samples: Rule: a(b+c) = ab+bc You can distribute the coefficient through the parenthesis with multiplication and remove the parenthesis.Rule:a(b+c) = ab+bcDiscuss/illustrate how arrows can help a student stay on trackSamples:4(3+2)=4(3)+4(2)=12+8=202(x+3) = 2x + 6-(3+x) = -3 - x
22 All real numbers have closure. Closure PropertyAll real numbers have closure.The Closure property states the if a and bare real numbers then:a + b is a real numberab is a real number.So, if you add two rational numbers, your sumwill be rational. Also, if you add two irrationalnumbers, that sum will be irrational.