Mathematics
Linear Expressions
Linear expressions are algebraic expressions that involve variables raised to the power of 1. They can be written in the form ax + b, where a and b are constants and x is the variable. These expressions represent lines on a graph and have a constant rate of change.
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8 Key excerpts on "Linear Expressions"
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)
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 . Objectives 1 Differentiate between an Expression and an Equation 2 Translate English Sentences into Mathematical Equations 3 Identify Strict Inequalities 4 Classify Inclusive Inequalities 5 Determine Possible Variable Values Based on Inequalities and the Phrases “At Most” and “At Least” 6 Determine Possible Variable Values for Compound Inequalities ■ ■ A variable is a letter used to represent an unknown quantity. ■ ■ A coefficient is a real number that is multiplied by a variable. ■ ■ An algebraic term is the product or quotient of a variable and a coefficient. ■ ■ An algebraic expression is the sum or difference of algebraic terms. Vocabulary An equation is a mathematical statement in which one algebraic expression is set equal to a constant or another algebraic expression. Equation As an example, in the algebraic expression 4 2 3.5 x x is the variable and the number 2 3.5 is the coefficient. The number 4 is not multiplied by a variable, so we call it a constant . If we set one algebraic expression equal to a constant or another algebraic expression, the result is an equation . As a result we determine that 4 2 3.5 x 5 2 31 is an equation, as is 4 2 3.5 x 5 15.5 1 6.1 x . In Example 1, we practice differentiating between an algebraic expression and an equation. EXAMPLE 1 Differentiating between an Algebraic Expression and an Equation Classify each of the following as an algebraic expression or an equation. Then identify the variables, coefficients, and constants. a. 1.7 5 35.8 1 z (11.2) b. s 21.5 SECTION 2.1 • Translating English to Algebra: Expressions, Equations, and Inequalities 57 Perform the Mathematics a.
eBook - ePubCommon Core Standards for Middle School Mathematics
A Quick-Start Guide
- Amitra Schwols, Kathleen Dempsey, John Kendall(Authors)
- 2013(Publication Date)
- ASCD(Publisher)
Chapter 4Expressions and Equations
. . . . . . . . . . . . . . . . . . . .The introduction to the Common Core standards' high school Algebra domain provides some useful definitions to help differentiate the two mathematical terms expression and equation:An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function…. An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. (CCSSI, 2010c, p. 62)The middle school Expressions and Equations (EE) domain provides a critical bridge between content in the Operations and Algebraic Thinking domain in earlier grades and algebraic content students will encounter in high school. Figure 4.1 shows an overview of the Expressions and Equations domain's clusters and standards by grade level.
- 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 - PDF- Kris Jamsa(Author)
- 2023(Publication Date)
- For Dummies(Publisher)
The following expressions are linear.” y x y x y x 1 2 3 Say to them: “Because of their use of exponents, the following expressions are not linear.” y x y x y x 2 3 3 2 1 This book’s companion website at www.dummies.com/go/teachingyourkids newmath6-8fd contains a worksheet (the first few rows of which are shown in Figure 26-1) that your child can use to identify an expression as linear or nonlinear. Download and print the worksheet. Help your child solve the first few problems and then ask them to complete the rest. FIGURE 26-1: A worksheet to identify Linear Expressions. CHAPTER 26 Calculating a Line’s Slope and Intercept 341 Understanding the Slope and Intercept of a Linear Equation When you look at a graph of a linear equation, the line may go up, down, or across the chart: The slope of a graph tells you the direction the graph is headed as x values increase. In this section, you teach your child how to calculate a line’s slope. Say to your child: “When you graph a linear equation, you use the term slope to describe the direction of the line. As x values increase, a positive slope goes up and a negative slope goes down.” Explain to them: “The slope of a line is a number that indicates how far the line moves up (or down) as it moves across the page. The following graphs have lines with different slopes.” HOW WE USE y = mx + + b You may be wondering why we care about the equation of a line y mx b . It turns out, data analysts often use the equation to predict future values. Assume, for example, that you have data that tells you about your sales (y) based on the number of salespeo- ple (x). When you plot the data, you calculate the slope of your line is 4 and your y- intercept (sales with 0 salespeople) is 5: y x or sales times the number of salespeople 4 5 4 5. Using the expression, you can predict your future sales if you were to add additional salespeople.
eBook - PDFFinite Mathematics
Models and Applications
- Carla C. Morris, Robert M. Stark(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
10 LINEAR EQUATIONS AND MATHEMATICAL CONCEPTS Slope Slope = m = Rise Run = Change in y Change in x = y x = y 2 − y 1 x 2 − x 1 The slope-intercept form of a line is y = mx + b, where m is the slope and b its y-intercept. A horizontal line has a slope of zero. A vertical line has an infinite (undefined) slope, as there is no change in x for any change in y. y y x x Positive slope Negative slope y y x x Undefined slope Zero slope Slope-Intercept Form of a Linear Equation y = mx + b where m is the slope and b the y-intercept. A linear equation in standard form (ax + by = c) is written in slope-intercept form by solving for y. Example 1.2.3 Slope-Intercept Form Write 2x + 3y = 6 in slope-intercept form and identify the slope and y-intercept. Solution: Solving for y, y = (−2∕3)x + 2. By inspection, the slope is −2∕3 (“line falls”) and the y-intercept is (0, 2), in agreement with the previous example. EQUATIONS OF LINES AND THEIR GRAPHS 11 A linear equation can also be written in a point-slope form: y − y 1 = m(x − x 1 ). Here, (x 1 , y 1 ) is a point on the line and m is the slope. Point-Slope Form of a Linear Equation y − y 1 = m(x − x 1 ) where m is the slope and (x 1 , y 1 ) a point on the line. Example 1.2.4 Point-Slope Form Find the equation of a line passing through (2, 4) and (5, 13) in point-slope form. Solution: First, the slope m = 13 − 4 5 − 2 = 9 3 = 3. Now, using (2, 4) in the point-slope form, we have y − 4 = 3(x − 2). (Using the point (5, 13) yields the equivalent y − 13 = 3(x − 5).) For the slope-intercept form, solving for y yields y = 3x − 2. The standard form is 3x − y = 2. Incidentally, as noted earlier, while the generic symbols x and y are most common, other letters are used to denote variables. Earlier I = prt was used to express accrued interest. Economists use q and p for quantity and price, respectively, and scientists often use F and C for Fahrenheit and Celsius temperatures or p and v for gas pressure and volume, and so on.
- S. Graham Kelly(Author)
- 2008(Publication Date)
- CRC Press(Publisher)
65 Chapter 2 Linear algebra 2.1 Introduction Linear algebra is the algebra used for analysis of linear systems. It provides a foundation on which solutions to mathematical problems can be developed. Success in obtaining a solution to a mathematical problem requires finding the specific solution among a possible set of solutions, the solution space. An understanding of the solution space and the properties of elements of the solution space leads to the development of solution techniques. It is in this spirit that this review of linear algebra is presented. An exact solution of a problem is a solution for the dependent variables which satisfies without error the mathematical problem for all possible values of the independent variables. An exact solution, while desirable, is not always possible. Approximate solutions are sought when an exact solution is not available. Approximate solutions are of two types. Variational methods are used to determine continuous functions of the independent variables which provide in some sense the “best approximation,” chosen from a specified set, to the exact solution. Numerical solutions provide an approximation to the exact solution only at discrete values of independent variables. Linear algebra provides a framework in which these approximate solutions can be devel-oped and in which the error between the exact solution and an approximate solution can be estimated. Linear algebra provides a framework for developing solutions to linear prob-lems. Modeling of engineering systems often leads to nonlinear mathematical problems. Exact solutions exist for only a few nonlinear problems. Often assump-tions are made such that the nonlinear problem can be approximated by a linear problem. Even if the assumptions that linearize the problem are not valid, some understanding of the solution can be obtained by studying the linearized prob-lem.
eBook - PDFAlgebra
Form and Function
- William G. McCallum, Eric Connally, Deborah Hughes-Hallett(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The choice of which letters to treat as variables and which as constants depends upon the context. 54 Chapter 2 LINEAR FUNCTIONS Exercises and Problems for Section 2.2 IDENTIFYING ALGEBRAIC STRUCTURE In Exercises 1–6, identify the constant term and coefficient in the expression + for the linear functions. 1. () = 200 + 14 2. () = 77 − 46 3. ℎ() = ∕3 4. () = 0.003 5. () = 2 + 7 3 6. () = √ 7 − 0.3 √ 8 In Exercises 7–12, give the constant term and the coefficient of for each of the Linear Expressions. 7. 3 + 4 8. 5 − + 5 9. + + 1 10. + 11. + + 5 + + 7 12. 5 − 2( + 4) + 6(2 + 1) In Exercises 13–16, the form of the expression for the func- tion tells you a point on the graph and the slope of the graph. What are they? Sketch the graph. 13. () = 3( − 1) + 5 14. () = 4 − 2( + 2) 15. () = ( − 1)∕2 + 3 16. ℎ() = −5 − ( − 1) 17. For working hours a week, where ≥ 40, a personal trainer is paid, in dollars, () = 500 + 18.75( − 40). What is the practical meaning of the 500 and the 18.75? 18. When guests are staying in a room, where ≥ 2, the Happy Place Hotel charges, in dollars, () = 79 + 10( − 2). What is the practical meaning of the 79 and the 10? 19. A salesperson receives a weekly salary plus a commis- sion when the weekly sales exceed $1000. The person’s total income in dollars for weekly sales of dollars (where ≥ 1000) is given by () = 600 + 0.15( − 1000). What is the practical meaning of the 600 and the 0.15? 20. Ashley receives an MP3 player as a gift. The number of songs in her collection months after receiving the MP3 player is given by () = 500 + 19( − 24). (a) What is the practical interpretation of the constants 24 and 500 in the expression for ? (b) Express () in slope-intercept form and interpret the slope and intercept. 21. A company’s profit after months of operation is given by () = 1000 + 500( − 4).
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