Linear Expressions

3 Key excerpts on "Linear Expressions"

• 

  eBook - ePub

  Basic Mathematics

  • Lawrence A. Trivieri(Author)
  • 2011(Publication Date)
  • Collins Reference

    (Publisher)

  CHAPTER 11

  An Introduction to Algebra

  I n this chapter, we introduce algebra as a generalization of arithmetic. In algebra, we use letters to represent numbers. Algebraic expressions are discussed, including evaluating and combining them. Removing symbols of grouping is also discussed. In addition, the language of algebra is introduced. Linear equations in a single variable is also discussed, together with the applied problems they can help solve.

  11.1 EVALUATING ALGEBRAIC EXPRESSIONS In this section, you learn to classify algebraic expressions as sums or products and to evaluate an algebraic expression. ALGEBRAIC EXPRESSIONS

  In algebra, letters are used to represent numbers. Such letters are called variables. The numbers are called constants. For example, in the expression × + 3y, the constant is 3; the variables are x and y . An algebraic expression is an expression that consists of constants and variables. Operation symbols and grouping symbols may also be included in the expression.

  The following are algebraic expressions:

  • x + y 2 + z3
  • –2a +3bc
  • 5m 3 – 3q5
  • –7xyz
  • u 3 –8u + u +

  TERMS AND FACTORS

  For our purposes, a difference is treated as an algebraic sum. Hence, x – y means x + (–y ). The expression 3u – 2v is called a sum of two terms, which are 3u and – 2v .

  • The algebraic expression 4t 2 – 3t can be written as 4t 2 + (–3t ). The constants are 4, 2, and –3; the variable is t
  • The algebraic expression 3s 4 – 2s – 3 + can be written as 3s 4 + (–2s ) + (–3)
  • The constants are 3, –2, and – 3; the variable is s .

  In an algebraic expression, a term is part of a sum. Terms are separated by + or – signs. A factor is part of a product.

  Exercise 11.1

  Determine whether each of the following expressions is a sum or a product. If a sum, determine the number of terms; if a product, determine the number of factors.

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  eBook - ePub

  Finite Mathematics

  Models and Applications

  • Carla C. Morris, Robert M. Stark(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  It is a conditional equation since it is only true when. Equations that hold for all values of the variable are called identities. For example, is an identity. By solving an equation, values of the variables that satisfy the equation are determined. An equation in which only the first powers of variables appear is a linear equation. Every linear equation in a single variable can be solved using some or all of these properties: Substitution – Substituting one expression for an equivalent one does not alter the original equation. For example, is equivalent to or. Addition – Adding (or subtracting) a quantity to each side of an equation leaves it unchanged. For example, is equivalent to or. Multiplication – Multiplying (or dividing) each side of an equation by a nonzero quantity leaves it unchanged. For example, is equivalent to or. To Solve Single Variable Linear Equations 1. Resolve fractions. 2. Remove grouping symbols. 3. Use addition and/or subtraction to move variable terms to one side of the equation. 4. Divide the equation by the variable coefficient. 5. Verify the solution in the original equation as a check. Example 1.1.1 Solving a Linear Equation Solve. Solution: To remove fractions, multiply both sides of the equation by 6, the least common denominator of 2 and 3. The revised equation becomes Next, remove grouping symbols to yield Now, subtract 4x and add 48 to both sides to yield Finally, divide both sides by 5 (the coefficient of x) to attain. The result, is checked by substitution in the original equation: Equations often contain more than one variable. To solve linear equations in several variables simply bring the variable of interest to one side. Proceed as for a single variable, considering the other variables as constants for the moment. Example 1.1.2 Solving for y Solve for y :. Solution: Move terms with y to one side of the equation and any remaining terms to the opposite side. Here,

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  eBook - ePub

  Mathematics for Economics and Finance

  • Michael Harrison, Patrick Waldron(Authors)
  • 2011(Publication Date)
  • Routledge

    (Publisher)

  Systems of linear equations and matrices

  DOI: 10.4324/9780203829998-3

  1.1   Introduction

  This chapter focuses on matrices. It begins by discussing linear relationships and systems of linear equations, and then introduces the matrix concept as a tool for helping to handle and analyse such systems. Several examples of how matrices might arise in specific economic applications are given to motivate the mathematical detail that follows. These examples will be used again and further developed later in the book. The mathematical material that follows the examples comprises discussions of matrix operations, the rules of matrix algebra, and a taxonomy of special types of matrix encountered in economic and financial applications.

  1.2    Linear equations and examples

  Linear algebra is a body of mathematics that helps us to handle, analyse and solve systems of linear relationships. A great deal of economics and finance makes use of such linear relationships. A linear relationship may be represented by an equation of the form

  z = α x + β y

  1.1

  where x, y and z are variables and α and ß are constants. Such relationships have several nice properties. One is that they are homogeneous of degree one, or linearly homogeneous, i.e. if all variables on the right-hand side are scaled (multiplied) by a constant, θ, then the left-hand side is scaled in the same way. Specifically, using (1.1 ), we have

  z * = α

  (

  θ x

  )

  + β

  (

  θ y

  )

  = θ

  (

  α x + β y

  )

  = θ z

  1.2

  Another property of linear relationships is that, for different sets of values for their variables, they are additive and their sum is also linear. Suppose we have the two equations z1 = αx1 + ßy1 and z2 = αx2 + ßy2 , then

  z 1

  +

  z 2

  = α

  (

  x 1

  +

  x 2

  )

  + β

  (

  y 1

  +

  y 2

  )

  1.3 after slight rearrangement, which may be written as

  Z = α X + β Y

  1.4 where X = x1 + x2 , Y = y1 + y2 and Z = z1 + z2 . The result, equation (1.4) , is a linear equation in the sums of the respective variables. The generalization to the case of n equations is straightforward and has

  X =

  ∑

  i = 1

  n

  x i

  ,   Y

  ∑

  i = 1

  n

  y i

    and   Z =

  ∑

  i = 1

  n

  z i

  .

  These simple properties constitute one reason why linear relationships are so widely used in economics and finance, and particularly when relationships, such as demand and supply curves, are first introduced to students.1

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