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**Literal Equations, Numbers, and Quanties**

Module 1 Lesson 2

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**What is a Literal Equation?**

A Literal Equation is an equation with two or more 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.

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**Solving a Literal Equation**

Step 1 Locate the variable you are asked to solve for in the equation. Step 2 Identify the operations on this variable and the order in which they are applied. Step 3 Use inverse operations to undo operations and isolate the variable.

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**Example: Solving Literal Equations**

A. Solve x + y = 15 for x. x + y = 15 Since y is added to x, subtract y from both sides to undo the addition. –y –y x = –y + 15 B. Solve pq = x for q. pq = _x_ p p Since q is multiplied by p, divide both sides by p to undo the multiplication.

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**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.

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**Your Turn! Solve for the indicated variable. 1. 2. for h**

3. 2x + 7y = 14 for y 4. for h P = R – C for C for m 5. for C

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**Your Turn Solutions H = 3V A 2. y = 14 – 2x 7 3. C = R – P**

4. m = x(k – 6 ) 5. C = Rt + S

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Example: Application The 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 both sides by to undo the multiplication. Now use this formula and the information given in the problem.

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Example: Continued The 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.

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The Real Number System Every 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.

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**Examples of each Subset**

Numbers Examples

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**Consider the following set of numbers.**

Example Consider the following set of numbers. List the numbers in the set that are: Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real numbers

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**Consider the following set of numbers.**

Example Consider the following set of numbers. List the numbers in the set that are: Natural Numbers: √16 = 4, so that is the only Natural Number Whole Numbers: 0 , √16 Integers: -3, 0, √16 Rational Numbers: -3, 0, ½ , .95, √16 Irrational Numbers: √8 Real numbers: All of the numbers listed above!

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**Properties of Real Numbers**

Property Addition Multiplication Commutative a + b = b + a ab = ba Associative (a + b) + c = a + (b + c) (ab)c = a(bc) Identity a + 0 = a a * 1 = a Inverse a + (-a) = 0 a * 1/a = 1 Distributive a(b + c) = ab + ac

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**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 Addition a+b = b+a Commutative Property of Multiplication ab = ba Samples: Commutative Property of Addition 1+2 = 2+1 Commutative Property of Multiplication (2x3) = (3x2) The next slide will discuss how these do not apply to subtraction and division.

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**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.

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**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? ZERO What can you multiply a number by and get the number back? ONE Samples: Identity Property of Addition 3+0=3 Identity Property of Multiplication 2(1)=2 Rules: Identity Property of Addition a+0 = a Identity Property of Multiplication a(1) = a

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**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 Addition a+(-a) = 0 Inverse Property of Multiplication a(1/a) = 1 Samples: Inverse Property of Addition 3+(-3)=0 Inverse Property of Multiplication 2(1/2)=1

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**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+bc Discuss/illustrate how arrows can help a student stay on track Samples: 4(3+2)=4(3)+4(2)=12+8=20 2(x+3) = 2x + 6 -(3+x) = -3 - x

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**Identify the property that justifies each of the following.**

6 + 8 = 8 + 6 5 + (2 + 8) = (5 + 2) + 8 =12 5(2 + 9) = (5 2) + (5 9) 4 (8 2) = (4 8) 2 5/9 9/5 = 1 5 24 = 24 5 = 0 -34 1 = -34

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**Solutions Communitive Associative Identity Distributive Inverse**

Indentity

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**All real numbers have closure.**

Closure Property All real numbers have closure. The Closure property states the if a and b are real numbers then: a + b is a real number ab is a real number. So, if you add two rational numbers, your sum will be rational. Also, if you add two irrational numbers, that sum will be irrational.

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