Mathematics
Expressions and Formulas
Expressions in mathematics are combinations of numbers, variables, and operations, while formulas are specific types of expressions that represent relationships or rules. Expressions can be evaluated to produce a single value, while formulas are used to solve specific problems or calculate specific quantities. Both expressions and formulas are fundamental to mathematical problem-solving and modeling.
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eBook - ePub- Chris Nord(Author)
- 2021(Publication Date)
- Chemeketa Press(Publisher)
CHAPTER 4Expressions
What is a Number?
The English language is full of figures of speech — such as “beating around the bush” — that we use to explain ideas. These are phrases that have meaning beyond their literal, word-for-word interpretation. When we say we’re “beating around the bush,” we mean avoiding a major, important topic by addressing minor, unimportant topics instead. If someone asks us what we’re beating, we say something like “Oh, that’s just an expression,” a combination of words that work together to create meaning.In math, an expression is a combination of numbers, variables, and operation symbols that together have meaning. In math, the meaning of an expression is a number. In this chapter, we will distinguish betweenarithmetic expressions, which are expressions that don’t contain variables, andalgebraic expressions, which do contain variables.4.1 Introduction to Expressions
In this section, you will learn how to evaluate arithmetic and algebraic expressions. You will also learn how to represent and interpret measurements in scientific notation. Finally, we will officially introduce formulas and learn two important ones — the area formula for a rectangle and the area formula for a triangle.A. Arithmetic Expressions
An arithmetic expression is a combination of numbers and operation symbols. This can be as simple as a single number, such as 23, or it can be a much more complicated combination of numbers and operation symbols, such as .An arithmetic expression can always be evaluated by performing each of the operations correctly and in the correct order. Once an arithmetic expression has been evaluated, it can then be replaced with its number value. Both the original expression and its evaluated form have the same meaning.
eBook - PDF- Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
What is Algebra? Understanding the fundamental rules of algebra is important in order to calculate results for a wide variety of situations in science and technology. Algebra is a language of mathematics that is based on generalized arithmetic using letters to represent numbers in expressions, equations, or formulas. 2 ⋅ 2 ⋅ 2 ⋅ 2 = 2 4 translates to a ⋅ a ⋅ a ⋅ a = a 4 and more generally a a a a a a n n factors ( )( )( )( ) ( ) = The material in this chapter is the foundation on which other later chapters in this book are built, so your understanding of the concepts in this chapter is important. 2 ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to: • Identify and understand all the components (literal terms, constants, and variables) that make up algebra. • Add and subtract algebraic expressions and identify groups using symbols like parentheses and brackets. • Complete operations using integral exponents and remove groups of symbols in algebraic expressions. • Complete operations with algebraic expressions using multiplication. • Complete operations with algebraic expressions using division. Introduction to Algebra 2–1 Algebraic Expressions Mathematical Expressions A mathematical expression is a grouping of mathematical symbols, such as signs of operation, numbers, and letters. ◆◆◆ Example 1: The following are mathematical expressions: (a) x 2 - 2x + 3 (b) x 5 sin 3 ⋅ (c) + x e 5 log x 2 ◆◆◆ 41 Section 2–1 ◆ Algebraic Expressions Algebraic and Transcendental Expressions An algebraic expression is one containing algebraic symbols and operations (addition, subtrac- tion, multiplication, division, roots, and powers), such as in Example 1(a). A transcendental expression, such as in Examples 1(b) and (c), contains trigonometric, exponential, or logarith- mic functions. Equations None of the expressions in Example 1 contains an equal sign ( = ).
eBook - PDF- Donald Davis, William Armstrong, Mike McCraith, , Donald Davis, William Armstrong, Mike McCraith(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
55 r.nagy/Shutterstock.com Chapter 2 CHAPTER CONTENTS Section 2.1 Translating English to Algebra: Expressions, Equations, and Inequalities Section 2.2 Order of Operations and Evaluating Numerical Expressions Section 2.3 Basics of Solving Linear Equations Algebraic Expressions Used in Statistics and Basics of Solving Equations 56 CHAPTER 2 • Algebraic Expressions Used in Statistics and Basics of Solving Equations SECTION 2.1 Translating English to Algebra: Expressions, Equations, and Inequalities When most people think of arithmetic and algebra, what comes to mind is a page full of numbers and letters, with very few words or sentences associated with them. Statistics is a bit different. Many applications in statistics are based on situations that are presented in narrative form, with our task being to translate that narrative into mathematics. In this section, we learn how to translate English into algebra. OBJECTIVE 1 Differentiate between an Expression and an Equation What differentiates arithmetic from algebra is that algebra is the part of mathematics in which letters are used to represent numbers that are either unknown or change from time to time. These letters are known as variables . Commonly used variables include x , y , z , n , t , and so on. In statistics, some variables are denoted by lower case Greek letters such as m (pronounced “mu”), s (pronounced “sigma”), l (pronounced “lambda”), and x (pronounced “kai”). If we place a real number in front or behind a variable, we call that number a coefficient . For our purposes, many of the coefficients will be represented by rational numbers. Together, variables and coefficients make up algebraic terms . Terms that are separated with addition and subtraction signs are called algebraic expressions .
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
244 Fundamentals of Algebra 4 UNIT 39 Symbolism and Algebraic Expressions OBJECTIVES After studying this unit you should be able to ● ● Express word statements as algebraic expressions. ● ● Express diagram dimensions as algebraic expressions. ● ● Evaluate algebraic expressions by substituting numbers for symbols. Algebra is a branch of mathematics in which letters are used to represent numbers. By the use of letters, general rules called formulas can be stated mathematically. Algebra is an extension of arithmetic; therefore, the rules and procedures that apply to arithmetic also apply to algebra. Many problems that are difficult or impossible to solve by arithmetic can be solved by algebra. The basic principles of algebra discussed in this text are intended to provide a practical background for machine shop applications. A knowledge of algebraic fundamentals is essential in the use of trade handbooks and for the solutions of many geometric and trigonometric problems. SYMBOLISM Symbols are the language of algebra. Both arithmetic numbers and literal numbers are used in algebra. Arithmetic numbers are numbers that have definite numerical values, such as 4, 5.17, and 7 8 . Literal numbers are letters that represent arithmetic numbers, such as a, x, V, and P . Depending on how it is used, a literal number can represent one particular arith-metic number, a wide range of numerical values, or all numerical values. Customarily, the multiplication sign s 3 d is not used in algebra, because it can be misin-terpreted as the letter x . When a literal number is multiplied by a numerical value, or when two or more literal numbers are multiplied, no sign of operation is required. SECTION FOUR UNIT 39 SYMBOLISM AND ALGEBRAIC EXPRESSIONS 245 Examples 1. 5 times a is written 5 a 2. 17 times c is written 17 c 3.
eBook - PDF- Dale Ewen(Author)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
For more information, please visit www.cengage.com and access the Student Online Resources for this text. CHAPTER 6 OBJECTIVES ◆ Use the addition, subtraction, multiplication, and division properties of equality to solve simple equations. ◆ Solve equations with parentheses. ◆ Solve equations with fractions. ◆ Translate words into algebraic symbols. ◆ Solve application problems using equations. ◆ Solve a formula for a given letter. ◆ Substitute data into a formula and find the value of the indicated letter using the rules for working with measurements. ◆ Substitute data into a formula involving reciprocals and find the value of the indicated letter using a scientific calculator. Equations and Formulas sima/Shutterstock.com Diesel Technician Diesel technician repairing a diesel engine 222 CHAPTER 6 ◆ Equations and Formulas Equations 6.1 In technical work, the ability to use equations and formulas is essential. A variable is a sym-bol (usually a letter of the alphabet) used to represent an unknown number. An algebraic expression is a combination of numbers, variables, symbols for operations (plus, minus, times, divide), and symbols for grouping (parentheses or a fraction bar). Examples of alge-braic expressions are 4 x 2 9, 3 x 2 1 6 x 1 9, 5 x (6 x 1 4), 2 x 1 5 2 3 x An equation is a statement that two quantities are equal. The symbol “ 5 ” is read “equals” and separates an equation into two parts: the left member and the right member. For example, in the equation 2 x 1 3 5 11 the left member is 2 x 1 3 and the right member is 11. Other examples of equations are x 2 5 5 6, 3 x 5 12, 4 m 1 9 5 3 m 2 2, x 2 2 4 5 3( x 1 1) To solve an equation means to find what number or numbers can replace the variable to make the equation a true statement. In the equation 2 x 1 3 5 11, the solution is 4. That is, when x is replaced by 4, the resulting equation is a true statement.
- Mark Zegarelli(Author)
- 2022(Publication Date)
- For Dummies(Publisher)
Any letter that you use to stand for a number is a variable, which means that its value can vary — that is, its value is uncertain. In contrast, a number in algebra is often called a constant because its value is fixed. Sometimes you have enough information to find out the identity of x. For example, consider the following: 2 2 x Obviously, in this equation, x stands for the number 4. But other times, what the number x stands for stays shrouded in mystery. For example: x > 5 In this inequality, x stands for some number greater than 5 — maybe 6, maybe 7 1 2 , maybe 542.002. Expressing Yourself with Algebraic Expressions In Chapter 6, I introduce you to arithmetic expressions: strings of numbers and operators that can be evaluated or placed on one side of an equation. For example: 2 3 + 7 1 5 2 . 2 4 400 4 − − − In this chapter, I introduce you to another type of mathematical expression: the algebraic expression. An algebraic expression is any string of mathematical symbols that can be placed on one side of an equation and that includes at least one variable. Here are a few examples of algebraic expressions: 5 x 5 2 x x y xy z xyz 2 2 3 1 As you can see, the difference between arithmetic and algebraic expressions is simply that an algebraic expression includes at least one variable. In this section, I show you how to work with algebraic expressions. First, I demonstrate how to evaluate an algebraic expression by substituting the values of its variables. Then I show you how to separate an algebraic expression into one or more terms, and I walk through how to identify the coefficient and the variable part of each term. CHAPTER 22 Working with Algebraic Expressions 467 Evaluating Algebraic Expressions To evaluate an algebraic expression, you need to know the numerical value of every variable. For each variable in the expression, substitute, or plug in, the number that it stands for and then evaluate the expression.
eBook - PDFAlgebra Teacher's Activities Kit
150 Activities that Support Algebra in the Common Core Math Standards, Grades 6-12
- Judith A. Muschla, Gary Robert Muschla, Erin Muschla-Berry(Authors)
- 2015(Publication Date)
- Jossey-Bass(Publisher)
3–2: (6.EE.2) WRITING AND READING ALGEBRAIC EXPRESSIONS For this activity, your students will be given phrases containing algebraic expressions. They are to determine if each expression is stated correctly. If it is incorrect, they are to correct the expression. Completing a statement at the end of the worksheet will enable students to check their work. Explain that algebraic expressions contain variables. A variable is a letter that represents a number. Discuss the examples on the worksheet. Emphasize that when writing an expression, order does not matter for addition and multiplication, but order does matter for subtraction and division. Grouping symbols indicate that a quantity must be treated as a unit. You might want to caution your students to be careful not to read the variable o as a zero. Review the directions on the worksheet. Students must correct the incorrect expressions and use the variables of the corrected expressions to complete the statement at the end. ANSWERS Answers to incorrect problems are provided. (2) s – 2 (5) ( t + 6 ) ÷ 12 (7) 4 2 + u (8) p ( 8 – 2 ) (11) ( e − 5 ) ÷ 6 (13) n – 15 (16) 8 d – 1 (17) 6 ( o + 3 ) (19) 3 u ÷ 3 (20) 22 s Your work with algebraic expressions is “stupendous.” 3–3: (6.EE.2) EVALUATING ALGEBRAIC EXPRESSIONS Your students will evaluate algebraic expressions in equations for this activity. By unscrambling the letters of their answers to find a math word, they can check their answers. Explain that an equation is a mathematical sentence that expresses a relationship between two quantities. Formulas are a special type of equation. Provide this example. The perimeter of a square is four times the length of a side. This can be written as an equation P = 4 s . Students can find the value of P if they know the length, s , of a side. If s = 12 . 5 inches, students can substitute 12.5 for s into the expression 4 s to find that P = 50 inches. Go over the directions on the worksheet with your students.
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
48 Module 2 • The four basic algebraic operations Adding and subtracting in algebra So far, we have added and subtracted numbers. We know that in algebra we use letters as placeholders. We have also learnt in Module 1 that a term has a coefficient, a coefficient has a sign, a term has a base with or without an exponent and the base can be raised to a power. –3 a 2 term sign coefficient to the power exponent or index base Pre-knowledge Definitions Algebraic expression An algebraic expression is one or more algebraic terms in a phrase. It can include variables , constants and operating symbols, such as plus and minus signs. It is only a phrase, not a whole sentence, so it does not include an equal sign . Example: 3 x 2 – 2 y + 7 xy + y 2 – 5 Term Terms are elements that are separated by a plus or a minus sign only . The example above has five terms: 3 x 2 , –2 y , 7 xy , y 2 and –5. The sign in front of a term belongs to that term. Terms could consist of variables and coefficients, or constants. Variable A variable is a letter or symbol that represents an unknown value. In algebraic expressions letters represent variables. These letters are actually numbers in disguise. In the expression 3 x 2 – 2 y + 7 xy + y 2 – 5 the variables are x and y . We call these letters var iables because the numbers they represent can vary : we can substitute one or more numbers for the letters in the expression. Coefficient A coefficient is the number, together with its sign, which is multiplied by the variable in an algebraic expression. In 3 x 2 – 2 y + 7 xy + y 2 – 5 the coefficient of the first term is 3 , the coefficient of the second term is – 2 , the coefficient of the third term is 7 and the coefficient of the fourth term is 1 . If a term consists of only variables, as in y 2 , its coefficient is 1. Constant A constant is a number that cannot change its value. Constants are the terms in the algebraic expression that contain only numbers: they are the terms without variables.
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