Mathematics
Variables in Algebra
In algebra, variables are symbols used to represent unknown or changing quantities. They are typically denoted by letters such as x, y, or z. Variables can be manipulated and combined using mathematical operations to solve equations and express relationships between quantities.
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9 Key excerpts on "Variables in Algebra"
eBook - PDF- Richard W. Hartel, D.B. Hyslop, D.B. Hyslop, T.A. Howell Jr.(Authors)
- 2008(Publication Date)
- CRC Press(Publisher)
chapter one Algebra Perhaps the easiest way to begin a discussion about algebraic equations and their components is to use examples to point out the various terms. The fol-lowing expressions will be used to demonstrate the different parts of an equation: y = 3 x − 7 (1.1) y = 2a x + 3b (1.2) An equation is a mathematical statement that can be read like a sentence. Equation (1.1) may be read as “ y equals 3 times x minus 7.” Since both sides of the equation are equivalent, one must always be very careful that any arithmetic operation performed on one side of the equation be performed on the other side as well. This will be stressed in later portions of this chapter. Equations consist of variables, constants, and arithmetic operators. Math-ematically, an equation relates different variables and constants; however, equations become more vital as one understands how physical parameters are linked together through them. . Variables and constants .. Variables Variables are so named because they are allowed to “vary,” and their value may assume different amounts at different times or situations. When one substitutes x = 1 into equation (1.1), y is calculated to be −4; however, when x = 3 is plugged into equation (1.1), one calculates y = 2. One can see that x and y are variables in this equation. Variables may be further categorized into independent and dependent variables. One might be able to logically deter-mine that the value for y in equations (1.1) and (1.2) is dependent on the value of x in the equations. That is, as x is independently varied, different values for y will be produced. In many situations, the manner in which the equation is written will dictate which variable is dependent and which is independent. Math concepts for food engineering Usually, a term written by itself on the left side of the equation is the depen-dent variable, while variables on the right side of the equal sign are indepen-dent variables.
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 - PDF- Alberto D. Yazon(Author)
- 2019(Publication Date)
- Arcler Press(Publisher)
The two members of the equation may be equated with the two weights that may be used on a balance scale. This analogy may be used to teach students who are beginning to learn about the equations. In the analogy of the scale, it may be understood that any kind of addition or subtraction on either side may bring about an equal alteration on the other side. If this does not happen, the scale may not be balanced. This same logic governs the solving of equations, in which to maintain the equality, the changes must be made accordingly. 2.3.1. Constants and Variables The statements in algebra comprise of two elements: • constant; and • variables. A constant is an element that maintains its value as same during a certain problem. A variable is an element that may have its value varying or changing, during the problem. The constants may be of two kinds: • fixed; and • arbitrary. Algebra and Basic Math 47 The fixed constants may be any number such as 4, 8. 13, 3/5, or 3.8. The value of fixed constants does not vary ever. For example, in the equation, 8x + 9 = 2, where the numbers 8, 9 and 2 are fixed constants. Arbitrary constants are those that vary with respect to the problems and may be given values that may not be alike in all the problems. Arbitrary constants may be denoted by letters that may generally be picked from the starting of the alphabet like a, b, c or d. For example, in the following equation ax + b = c, where the letters a, b, and c are the arbitrary constants. The expression ax + b = c may be indicating to a lot of linear equations. These equations may turn into fixed constant type, if a, b, and c are assigned some specific values like, for example, a = 8, b = 9 and c = 23, in which case the equation takes the form: 8x + 9 = 23 A variable, on the other hand, may possess one or more than one value in a conversation.
eBook - PDF- (Author)
- 2015(Publication Date)
- For Dummies(Publisher)
Part II Algebra is Part of Everything In this part . . . ✓ Start with the basics of algebra to lay the foundations for more advanced aspects. ✓ Get those variables under control — know how to add, subtract, multiply, divide and distribute them! ✓ Understand the workings of factorisation and finding the prime factor. ✓ Work with quadratic expressions and how they can be FOILed (and unFOILed). Chapter 6 Understanding the Basics of Algebra In This Chapter a Reminding yourself about the number basics a Working through the algebraic symbols, and communicative and associative properties a Powering up with exponents a Understanding some of the more advanced aspects of exponents I n a nutshell, algebra is a way of generalising arithmetic. Through the use of variables (letters representing numbers) and formulas or equations involving those variables, you solve problems. The problems may be in terms of practical applications, or they may be puzzles for the pure pleasure of the solving. Algebra uses positive and negative numbers, integers, fractions, operations, and symbols to analyse the relationships between values. In this chapter, you find some of the basics necessary to more easily find your way through the required processes and rules. Looking at the Basics: Numbers The different sets of numbers that algebra relies on are important because what they look like and how they behave can set the scene for particular situations or help to solve particular problems. Algebra uses different sets of numbers, in different circumstances. I describe the different types of numbers here. 112 Part II: Algebra is Part of Everything Really real numbers Real numbers are just what the name implies. In contrast to imaginary numbers, they represent real values — no pretend or make‐believe. Real numbers cover the gamut and can take on any form — fractions or whole numbers, decimal numbers that can go on forever and ever without end, positives and negatives.
eBook - PDF- Daniel Ashlock(Author)
- 2022(Publication Date)
- Springer(Publisher)
1 C H A P T E R 1 Review of Algebra This book is a text on calculus, structured to prepare students for applying calculus to the physical sciences. The first chapter has no calculus in it at all; it is here because many students manage to get to the university or college level without adequate skill in algebra, trigonometry, or geometry. We assume familiarity with the concept of variables like x and y that denote numbers whose value is not known. 1.1 SOLVING EQUATIONS An equation is an expression with an equals sign in it. For example: x D 3 is a very simple equation. It tells us that the value of the variable x is the number 3. There are a number of rules we can use to manipulate equations. What these rules do is change an equation into another equation that has the same meaning but a different form. The things we can do to an equation without changing its meaning include the following. • Add or subtract the same term from both sides. If that term is one that is already present in the equation, we may call this moving the term to the other side. When this hap- pens, the term changes sign, from positive to negative or negative to positive. For example: x 4 D 5 This is the original equation x D 5 C 4 Add 4 to both sides (move 4 to the other side) x D 9 Finish the arithmetic • Multiply or divide both sides by the same expression. 2 1. REVIEW OF ALGEBRA For example: 3x 4 D 5 This is the original equation 3x D 5 C 4 Add 4 to both sides (move 4 to the other side) 3x D 9 Finish the addition 3x 3 D 9 3 Divide both sides by 3 x D 3 Finish the arithmetic • Apply the same function or operation to both sides. For example: p x 2 D 5 This is the original equation p x 2 2 D 5 2 Square both sides x 2 D 25 Do the arithmetic x D 27 Move 2 to the other side Knowledge Box 1.1 The rules for solving equations include: 1. Adding or subtracting the same thing from both sides. 2. Moving a term to the other side; its sign changes. 3. Multiplying or dividing both sides by the same thing.
eBook - PDF- Tom Bassarear, Meg Moss(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
In each of the following examples, the variables have a slightly different meaning. On a separate piece of paper, write down what each of the variables means and then read on. . . . 5 30 sin cos tan 1 1 C d x x x x n n y kx ⋅ ⋅ p 5 5 5 5 5 DISCUSSION The first example is a formula. C and d stand for circumference and diameter, whose values vary according to the circle; however, the value of p (3.14) does not vary. The second example is generally called an equation. Although x is the variable, in this case its value is 6. The third example is an identity; it is true no matter what the value of x. The fourth example represents a property. In this case, n is used as a symbol to represent a property that is true for all numbers (except 0); the formal name of this property is the multi- plicative inverse property. The fifth example represents a family of functions in which the independent variable x ( ) and the dependent variable y ( ) are related in a certain way—in this case, a linear relationship. Table 6.3 illustrates some of the important differences in these examples. Table 6.3 Example What is usually called What the variables stand for p 5 C d Formula C and d are concrete quantities. p is a constant. 5 x 5 30 Equation x is the unknown, and we can solve for it. 5 ⋅ x x x sin cos tan Identity x is an argument of a function. 5 / ⋅ n n 1 (1 ) Property n is a symbol to represent a generalization. 5 y kx An equation of a func- tion of direct variation x is the independent variable, y is the dependent variable, and k is a constant. Section 6.2 Representing and Analyzing Mathematical Situations and Structures Using Algebraic Symbols 301 equal.” Can you see how this circular definition does not really define the equal sign since it requires that we already know what “equals” means? How can we define “equal sign” without using the word “equals”? Try it and then read on. A first-grade teacher presented this problem to her children: 8 1 4 5 1 7, and they all said 12.
eBook - PDF- G. M. Fikhtengol'ts, I. N. Sneddon(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
F. Engels emphasizes this fact in the following words: A turning point in mathematics was Descartes' variable quantity. Thanks to this, mathematics encompassed motion and dialectic, t F. Engels. Dialectic of Nature, ed. 1952, p. 37 (in Russian). [25] 26 2. FUNCTIONS OF ONE VARIABLE and owing to this the differential and integral calculus became at once necessary... 1 ' 15. The domain of variation of a variable quantity. In mathemat-ical analysis—providing we do not speak of its applications—by a variable quantity (or briefly a variable) we mean an abstract or numerical variable. It is denoted by a symbol (a letter, e.g. x) which is endowed with numerical values. The variable x is regarded as prescribed if the set 9C = {x} of values, which the variable can acquire, is indicated. This set is called the domain of variation of the variable x. In general, any numerical set may serve as the domain of variability of a variable. A constant quantity (briefly a constant) may conveniently be regarded as a particular case of the variable: it corresponds to the assumption that the set 9C = {x} contains only one element. We found in Sec. 13 that the numbers have a geometric interpreta-tion as points on an axis. The domain 9C of variation of the varia-ble x is represented on this axis as a set of points. Accordingly, usu-ally the numerical values of the variable themselves are called points. Frequently we have to consider a variable n taking all possible positive integral values 1,2,3,... 100,101,...; the domain of variation of this variable, i.e. the set {«} of positive integers, will always be denoted by 9£. However, analysis is usually concerned with variables which vary in a continuous manner: they are derived from physical quan-tities—time, distance covered by a moving point, etc. The domain of variation of such a variable is a numerical interval.
- Helen J. Forgasz, Joanne Rossi Becker, Kyeonghwa Lee(Authors)
- 2016(Publication Date)
- Information Age Publishing(Publisher)
Gender Differences When Working with Algebraic Variables 193 The difficulty might be related to the multiple appearance of the vari- able as well as to the necessity to shift between viewing it as a general num- ber that should be manipulated without knowing its value and viewing the variable as an unknown whose value has to be determined. Another of the most common answers to this question was 3 or 3 values (see Figures 8.9 and 8.10); this answer may be related to assignation of the value (Küchemann, 1981) of the unknown quantity depending on the number of times it appears in the expression. We do not have evidence concerning their interpretation of the three quantities as the same quantity or as different quantities. Figure 8.9 Response of 3. Figure 8.10 Response of 3. In the same question two students (one male and one female) answered 3a = 10a showing a concatenation of the elements of the expression (see Figure 8.11). This shows students’ difficulty in making sense of and oper- ating with a variable. One boy who interpreted the variable as a general number contemplated negative numbers to indicate the symbolic repre- sentation of any quantity (see Figure 8.12). In contrast, one girl tried to manipulate the variable, considering it as a general number, although not appropriately (see Figure 8.13). Figure 8.11 Concatenation. Figure 8.12 Sample of negative number use. Figure 8.13 Use of variable as a general number. 194 C. ORTEGA and S. URSINI In exercise 5d (x + 5 = x + x) about the interpretation of variable, the most frequent answer was to interpret the variable only as a general num- ber without interpreting it as an unknown quantity. (Also we observed the concatenation of the elements of the equation obtained 5x = 2x females and males and the sameness was ignored.) It is possible that they consid- ered different quantities for each letter in the same expression or 3 values because the variable appears 3 times.
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.
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