Module 3 Basic Algebra.docx - Module 3 Basic Algebra Download Module as PDF Module_3_Basic_Alge bra Module 3 Learning Objectives After completing this

Module 3 Basic Algebra.docx - Module 3 Basic Algebra...

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Unformatted text preview: Module 3: Basic Algebra Download Module as PDF Module_3_Basic_Alge bra.pdf Module 3: Learning Objectives After completing this module, you should be able to: 1. 2. 3. 4. 5. Identify the inverse operation that corresponds to a given operation Evaluate a given algebraic expression using the distributive property Combine like terms to simplify a given algebraic expression Evaluate a given algebraic expression using substitution Identify an algebraic expression or equation that defines a given pattern of relationships for a given real world problem 6. Solve a given linear equation of a specified form 7. Predict solution to a real world problem given an algebraic equation 8. Solve a given linear equation with fraction coefficients and constants of the form (a/b)x=c/d 9. Solve a given linear inequality 10. Identify a graphical representation of the solution 11. Identify the graph of given coordinates on a coordinate plane 12. Identify the slope when given a graph of a linear equation 13. Identify the correct graph for a given linear equation of the form y=mx+b Algebraic Conventions and Notation Before we can enjoy the fruits of algebra, we must first master the conventions that are really at the heart of it. So much of algebra involves working with different values, manipulating numbers, and finding solutions. In the Number Systems module, we broke down numbers into their most basic parts. The conventions and notations within algebra do the same thing — they help us better understand the language of algebra. Arithmetic Terminology One may assume terminology is a trivial part of learning mathematics, however it is crucial when it comes to solving algebraic problems. To understand arithmetic terminology is to understand all the different, small pieces that make up the big picture. Operations You may remember learning how to add, subtract, multiply, and divide in grade school. While these concepts may seem basic, an understanding of algebra requires the ability to strictly define and master these concepts, known as operations. An operation is a mathematical procedure which can generate a new value. Elementary operations are the simplest and most common operations: addition, subtraction, multiplication, and division. Constants The word constant may seem intimidating at first, but you work with constants all of the time. A constant is a number with a fixed value. All real numbers are constants, including 0, 1.5, −10, and π. Any of the numbers we used in "Number Systems" are constants, regardless of whether they are prime or composite, odd or even, rational or irrational! Arithmetic Expressions An arithmetic expression is a string of numbers connected by elementary operations. Arithmetic expressions are the addition, subtraction, multiplication, and division problems you are probably familiar with. These expressions can be of any length. They can be short and simple: Or, an arithmetic expression can be longer: Exponents An exponent, sometimes called a power, is a quantity that represents repeated multiplication. An exponent is written above and to the right of a number, known as a base. The exponent's value is equal to the number of times the base is multiplied by itself. To the right, the base is 6 and the exponent 3. 63 is equal to 6×6×6 Exercise: Algebraic Conventions and Notation Answer the following questions by entering in the letter that corresponds with your answer choice. 1. Which of the following operations are elementary operations? a. b. c. d. D t Only addition and subtraction Only multiplication and division Only addition and multiplication Addition, subtraction, multiplication, and division 2. What elementary operation will you use to generate the value the constants 33 and 44? Rese 77 from A Rese t a. b. c. d. Addition Subtraction Multiplication Division 3. What elementary operation will you use to generate the value the constants 33 and 33? a. b. c. d. C 00 from B Rese t Addition Subtraction Multiplication Division 5. Which of the below are constants? Select the most correct answer. a. 3.23.2 b. 3737 c. −0.0009-0.0009 d. All of the above D −0.000016-0.000016 xx 3y3y 2w2w Rese t 6. Which of the following is a constant? a. b. c. d. Rese t Addition Subtraction Multiplication Division 4. What elementary operation will you use to generate the value the constants 66 and 66? a. b. c. d. 99 from A Rese t 7. In the expression 32×5÷1−232×5÷1-2, which value is a base? A t Rese a. b. c. d. 33 22 55 11 8. Which of the following contains an elementary operation? a. b. c. d. 16−−√16 4+54+5 1616 0.00000030.0000003 B t 9. Which of the following is an arithmetic expression? a. b. c. d. x=yx=y xx 55 2+182+18 D Rese t 10. What is 55 called in 125125? B t a. b. c. d. Rese a base an exponent an expression a variable Review Checkpoint To test your understanding of the content presented in this assignment, please click on the Question icon below. Click your selected response to see feedback displayed below it. If you have trouble answering, you are always free to return to this or any assignment to re-read the material. Rese 1.52 is equal to... a. 2×2×2×2×2 b. 5×5 Correct. The answer is b. The exponent's value is equal to the number of times the base is multiplied by itself. Here, the exponent is 2 and the base is 5. Therefore, 5 is multiplied by itself 2 times. c. 5×5×5 d. 2×5 2. True or False? All constants are rational numbers. a. True b. False Correct. This is a false statement. A constant can be rational or irrational. A constant is any number with a fixed value. Read all the options below before answering. 3. Which of the following is considered an arithmetic expression: a. −4+−4 b. 2−1×5 c. 5×(7/3)−9 d. −2×(−3+9−2)+5 e. All of the above Correct. The answer is e. All of the above are arithmetic expressions. They are each a string of numbers connected by elementary operations. 4. What is 74 is equal to? a. 4×4×4×4×4×4×4 b. 7×7×7×7 Correct. The answer is b. The exponent's value is equal to the number of times the base is multiplied by itself. Here, the exponent is 4 and the base is 7. Therefore, 7 is multiplied by itself 4 times. c. 7×4 d. 4×7×4×7 5. The following string of numbers and operations is an arithmetic expression. True or False? 10−5÷2×8+36 a. True Correct. This is a true statement. An arithmetic expression is a string of numbers connected by elementary operations, which are addition, subtraction, multiplication, and division. The given string of numbers is connected by elementary operations, therefore it is an arithmetic expression. b. False Understanding Variables For many, the word "mathematics" conjures up images of arithmetic. Those who are familiar with arithmetic, but unfamiliar with algebra, may only consider numbers, or constants, to be the language of mathematics. This notion overlooks the important role variables play in the language of mathematics. In elementary algebra, a variable is a symbol that represents or holds the place of a numerical value. Often, a variable will be a letter from the Western alphabet (a, b, c, …) or Greek alphabet (α, β, γ, …). The actual letter or symbol being used as a variable is not important. The numerical value represented by the symbol is what gives the variable its importance. Substitution When we know the value that a variable represents, we can simply replace the variable with its value. Then, we can evaluate the expression. For example, if we know that variable aa represents a value of 1, we can say that a equals 1. Because we know that the value of a is 1, we can replace the variable a with the number 1 in an expression. This is outlined in the examples below: Example #1 Va r i a b l e Va l u e a 1 Using the table above, we have a variable, aa, which represents the numerical value of 1. Therefore, the statement (a+5) is equivalent to (1+5). Because the variable aa represents the numerical value of 1, we can simply replace the variable aa with the number 1. Example #2 Examine another table of variables, and their values, below: Va r i a b l e Va l u e b 1 C 3 d 2 Using the table above, we can substitute any variable with the numerical value it represents. For example: b is equivalent to 1. C is equivalent to 3. d+4 is equivalent to 2+4. b−5 is equivalent to 1−5. C/2 is equivalent to 3/2. Exercise U s i n g Va r i a b l e s The following is another table of variables, and their values: Va r i a b l e Va l u e t 1 E 4 a 5 Using the table above, write out, but do NOT evaluate the expressions below. For example for the expression t+5, you would write 1+5 but not 6. Leave spaces between the numbers and operations as shown above (1+5 not 1+5). Use the keyboard x for multiplication and the / for division. 1. t is equivalent to 1 Reset 2. 1+E is equivalent to 1+4 3. a−2 is equivalent to 5-2 4. 6×t is equivalent to 5. E / 4 is equivalent to 6 X1 4/4 Reset Reset Reset Reset 6. E+2×E is equivalent to 7. a×a is equivalent to 5 X5 8. a / t is equivalent to 5/1 9. E×t is equivalent to 4 X1 10. a−E is equivalent to 5-4 4 + 2 X4 .Reset .Reset .Reset ·Reset .Reset Review Checkpoint To test your understanding of the content presented in this assignment, please click on the Question icon below. Click your selected response to see feedback displayed below it. If you have trouble answering, you are always free to return to this or any assignment to re-read the material. 1. You know that the variable x represents the value of 5, and the variable y represents the value of 3. Evaluate x+y. a. 2 b. 5 c. 8 Correct. The answer is c. Because the variable x represents the value of 5, we simply replace it with the number 5. Similarly, we replace the variable y with the number 3. Therefore, the statement x+y is equivalent to 5+3, which evaluates to 8. d. 15 2. You know that the variable aa represents the value of −2, and the variable bb represents the value of 7. Evaluate a⋅b. a. −14 Correct. The answer is a. Because the variable a represents the value of −2, we simply replace it with the number −2. Similarly, we replace the variable b with the number 7. Therefore, the statement a⋅b is equivalent to −2⋅7, which evaluates to −14. b. −5 c. 3 d. 9 3. You know that the variable L represents the value of −1, the variable M represents the value of −2, and the variable N represents the value of −3. Evaluate L+M+N. a. −12 b. −6 Correct. The answer is b. Because the variable L represents the value of −1, we simply replace it with the number −1. Similarly, we replace the variable M with the number −2, and the variable N with the number −3. Therefore, the statement L+M+N is equivalent to −1+(−2)+(−3), which evaluates to −6. c. −1 d. 3 4. You know that the variable p represents the value of 25, and the variable q represents the value of 5. Evaluate p÷q−5 a. −5 b. 0 Correct. The answer is b. Because the variable p represents the value of 25, we simply replace it with the number 25. Similarly, we replace the variable q with the number 5. Therefore, the statement p÷q−5 is equivalent to 25÷5−5, which evaluates to 0. c. 5 d. 12.5 5. The following expression contains variables. True or false? 10−5÷2×8+36 a. True b. False Correct. This is a false statement. The expression above does not contain an unknown symbol or letter which is a variable. Algebraic Expressions Career Connections Algebraic Expressions Algebraic expressions are useful whenever you want to perform a calculation multiple times. Algebra is a tool that multiplies the power of our mental capacity the way a bicycle multiplies the power of our physical capacity. You can do more, and more complex, math with algebra than you can on your own. Some algebraic expressions are tools that have already been developed. Businesses make use of many standard formulas, which are algebraic expressions. You might, for example, need to calculate the future value of money in order to determine the benefit of taking in money in the current fiscal year versus deferring that income to a later year. The future value is a standard formula that many businesses use. FV=PV×(1+r)n Where: FV=future value of money PV=present value of money r=interest rate n=number of periods You can draw on the formula and make different inputs every time you need to calculate the future value of a current income. This module will teach you how to input the values associated with these kinds of formulas. For example, a landscaping company can use an algebraic expression to calculate the price to charge customers to install plants. The formula will account for shrubs with varying wholesale prices and includes the percent markup plus labor costs. In this example, the number of each type of shrub will be multiplied by its wholesale cost, then multiplied by 120% for markup; plus, an additional $19.50 per hour per worker, although the actual salary is $15 per hour. Wholesale Plant Prices $10 Arctostaphylos uva-ursi (bearberry) Clethra alnifolia (sweet pepperbush) $8.5 0 Daphne 'Carol Mackie' $54 Ilex glabra $32 Kalmia latifolia $42 Morella pensylvanica $10 The retail cost for the customer of installing any combination of these native shrubs is given by the algebraic expression: 1.2(10A+8.5C+54D+32I+42K+10M)+19.5H In the expression, "A" represents the number of Arctostaphylos uva-ursi plants; " C" represents the number of Clethra alnifolia plants; " D" represents the number of Daphne 'Carol Mackie' plants, etc. "H" represents the number of hours needed for the installation. The expression can be used for any combination of plantings to quickly generate a quote. When the price for one of the shrubs changes, the expression can be easily updated to incorporate the new information. If labor costs go up, the number in front of the H can be increased. Moreover, it would be easy to modify this expression to calculate your own costs. Combining what you are charging the customer and what an installation costs you, would give you your profit on any particular job since: Profit=Revenue–Expense "Revenue" is what the customer is paying you and "expense" is what you are paying to the wholesale nursery for the plants and to your workers for their labor. Subtracting the latter from the former gives you your profit on each job. As the example above shows, algebraic expressions are a way of storing and mobilizing information. They provide a quick and accurate way to aggregate information. Expressions allow us to perform the same calculation repeatedly with minimal effort. We just change the inputs to account for the new situation. Algebraic expressions also let us perform calculations that are too complicated to be done in our heads. With our knowledge of variables and arithmetic terminology, we can begin to understand the pieces that make up the very crux of algebra. Those pieces include coefficients, terms, and expressions. Creating Terms: Multiplication Click through the slideshow below to learn more about multiplying variables. Multiplying Variables Variables are simply symbols that represent numerical values. While these variables represent a value, we don't always know which value it represents. When multiplying variables, we remember that variables do in fact represent values, just like constants. We can perform operations on variables without knowing their true values. Multiplying Variables Operations work similarly with variables and constants. Multiplication generates a product: a completely different operation than addition or subtraction. Take 2⋅6 for example. If we add 2 and 6, the sum is 8; but when multiplying 2 and 6, the product is 12. We're connecting these constants using multiplication in a different way than addition. As with numbers, multiplication of variables is a wholly unique operation. For Example: a⋅x With multiplication, these variables form a product. Our product is a⋅x. Why Multiplication is Unique Multiplication is the only elementary operation that can be denoted by variables and constants simply being written next to one another: 3⋅x can be written as 3x3x. In other words 3⋅x and 3x mean the same thing. Similarly, −6⋅a can be written as −6a, and 14⋅d can be written as 14d. If variables are being multiplied by one another, the same notation applies: a⋅b can be written as ab, for example. Addition, subtraction, and division all require some type of sign ( +,−,÷) to denote that they're being performed. Not multiplication. That's why we can connect two or more variables when multiplying them; simply being next to one another tells us that they're being multiplied. Example: d⋅m = dm The product is written as simply dm. We don't know the values of d and m, but seeing the variables directly next to one another tells us they're being multiplied. Example: g⋅y = gy The product is gy. Because the two variables are being multiplied, we can write them next to one another. Example: x⋅w⋅z = xwz This notation can be used with any number of variables! Exponents How about operations like a · a that involve two of the same variables being multiplied? You may recognize this operation as repeated multiplication. This type of operation is best represented using exponents. Exponents are used for repeated multiplication of a variable, the same as with repeated multiplication of a constant. a⋅a = a2 This is written as a2 (aa squared). As with constants, the number of times the variable is multiplied by itself determines the value of the exponent. Example: d⋅d⋅d = d3 Because d is being multiplied by itself 3 times, the exponent is 3 ("d cubed"). Example: x⋅y⋅y = xy2 Above, x and y are connected because they are being multiplied. Because y is being multiplied twice, it becomes y2. The product is written as xy2 ("xy squared"). Example: h⋅h⋅g =h 2 g Above, h is being multiplied twice, and of 2 ("h squared g"). g only once, so h has an exponent Coefficients In algebra, constants often take the form of coefficients. A coefficient is a number by which a variable is being multiplied. Coefficients are written in front of variables. So, in 16x, 16 is the coefficient and x is the variable. If a variable is without a number in front of it, the coefficient is 1. Though it is not written, there is essentially an invisible 1 in front of any variable without a numerical coefficient. Terms One of the fundamental building blocks of algebra is a term. A term can be many things: a single constant, such as 5, a single variable, such as x, or a term can also be any number of constants and variables multiplied together, such as 7ab. So, terms can include: Multiplication Constants, which can be coefficients Exponents Variables Writing variables and numbers next to one another indicates multiplication. So, 15xy2 is equivalent to 15⋅x⋅y⋅y. Only multiplication is denoted by numbers and variables being next to one another — other operations like addition, subtraction, and division require their corresponding operators. Though the expression to the right appears to be just one entity, it is, in fact, an expression made up of many terms. Each of the terms are circled in blue and are separated by either addition, subtraction, or division operations. When deciphering terms within an expression, be mindful of the operators that break each of them up. Algebraic Expressions An algebraic expression is a string of terms connected by division, addition, and subtraction. Consider the expression below. We recognize the individual terms that are separated from each other by division, addition and subtraction. These terms make up an algebraic expression. Below are a few different examples of algebraic expressions: (10÷2)−22 2x2+9 6a2b−π 4w3z−wz+8 Often, throughout this course, we will refer to algebraic expressions simply as "expressions." Writing Algebraic Expressions When writing an algebraic expression, you want to consider the order in which you write the terms that make up the expression. Write the constants at the end of your expression. 6x+11 Notice that the expression above has two terms: 6x6x and 11. One of these terms (11) is a constant. Because th...
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