1. BUS-221 Quantitative Methods
LECTURE 3

2. Learning Outcome
Knowledge - Be familiar with basic mathematical techniques including: systems of linear equations
Research - Retrieve and analyse information from directed sources for calculation and interpretation
 Mentation - Analyse business case studies and make decisions based on quantitative data.

3. Topics Functions ▪ common functions: ln, exp, sin, cos, polynomial
Linear equation systems

4. Equations, in Particular Linear Equations (1 of 12)
An equation is a statement that two expressions are equal.
The two expressions that make up an equation are called its sides.
They are separated by the equality sign, =.

5. Equations, in Particular Linear Equations (2 of 12)
Example 1 – Examples of Equations
A variable is a symbol that can be replaced by any one of a set of different numbers.
The most popular symbols are letters from the latter part of the alphabet, such as x, y, z, w and t.

6. Equations, in Particular Linear Equations (3 of 12)
Equivalent Equations
Two equations are said to be equivalent if they have exactly the same solutions.
There are three operations that guarantee equivalence:
Adding (subtracting) the same polynomial to (from) both sides of an equation.
Multiplying (dividing) both sides of an equation by the same nonzero constant.
Replacing either side of an equation by an equal expression.

7. Equations, in Particular Linear Equations (4 of 12)
Operations That May Not Produce Equivalent
Equations
Multiplying both sides of an equation by an expression involving the variable.
Dividing both sides of an equation by an expression involving the variable.
Raising both sides of an equation to equal powers.

8. Equations, in Particular Linear Equations (5 of 12)
A linear equation is also called a first-degree equation or an equation of degree one.

9. Equations, in Particular Linear Equations (6 of 12)
Example 2 – Solving a Linear Equation

10. Equations, in Particular Linear Equations (7 of 12)
Example 3 – Solving a Linear Equations

11. Equations, in Particular Linear Equations (8 of 12)
Literal Equations
Equations in which some of the constants are not specified, but are represented by letters such as a, b, c, or d, are called literal equations.
The letters are called literal constants.

12. Equations, in Particular Linear Equations (9 of 12)
Example 4 – Solving a Literal Equation

13. Equations, in Particular Linear Equations (10 of 12)
Fractional Equations
A fractional equation is an equation in which an unknown is in a denominator.
Example 9 – Solving a Fractional Equation

14. Equations, in Particular Linear Equations (11 of 12)
Example 5 – Literal Equation

15. Equations, in Particular Linear Equations (12 of 12)
Radical Equations
A radical equation is one in which an unknown occurs within a radical expression.
Example 13 – Solving a Radical Equation

16. Lines (1 of 11)
Slope of a Line

17. Lines (2 of 11)
Example 1 – Price-Quantity Relationship

18. Lines (3 of 11)
Example 1 – Continued

19. Lines (4 of 11)
Equations of Lines

20. Lines (5 of 11)
Example 2 – Determining a Line from Two Points

21. Lines (6 of 11)
Example 5 – Find the Slope and y-Intercept of a Line

22. Lines (7 of 11)
Example 3 – Converting Forms of Equations of Lines

23. Lines (8 of 11)
Example 3 – Continued

24. Lines (9 of 11) Parallel and Perpendicular Lines
Two lines are parallel if and only if they have the same slope or are both vertical.
Moreover, any horizontal line and any vertical line are perpendicular to each other.

25. Lines (10 of 11)
Example 4 – Parallel and Perpendicular Lines

26. Lines (11 of 11)
Example 4 – Continued

27. Systems of Linear Equations (1 of 5)
Two-Variable Systems
There are three different linear systems:
There are two methods to solve simultaneous equations:
elimination by addition method
elimination by substitution method

28. Systems of Linear Equations (2 of 5)
Example 1 – Elimination-by-Addition Method
Use elimination by addition to solve the system

29. Systems of Linear Equations (3 of 5)
Example 2 – A Linear System with Infinitely Many Solutions

30. Systems of Linear Equations (4 of 5)
Example 3 – Solving a Three-Variable Linear System

31. Systems of Linear Equations (5 of 5)
Example 4 – Two-Parameter Family of Solutions

32. Systems (1 of 2)
A system of equations with at least one nonlinear equation is called a nonlinear system.
Example 1 – Solving a Nonlinear System

33. Nonlinear Systems (2 of 2)
Example 1 – Solving a Nonlinear System

34. Exponential Functions (1 of 9)
Rules for Exponents

35. Exponential Functions (2 of 9)
Example 1 – Bacteria Growth

36. Exponential Functions (3 of 9)
Example – Graphing Exponential Functions with 0 < b < 1
Graph the exponential function
Solution:

37. Exponential Functions (4 of 9)
Table 4.1 Properties of the Exponential Function
1. The domain of any exponential function is (−∞, ∞).
The range of any exponential function is (0, ∞).
2. The graph of f(x) = bx has y-intercept (0,1).
There is no x-intercept.
3. If b > 1, the graph rises from left to right.
If 0 < b < 1, the graph falls from left to right.
4. If b > 1, the graph approaches the x-axis as x becomes more and more negative.
If 0 < b < 1, the graph approaches the x-axis as x becomes more and more positive.

38. Exponential Functions (5 of 9)
Example – Graph of a Function with a Constant Base
Solution:

39. Exponential Functions (6 of 9)
Exponential functions are involved in compound interest, whereby the interest earned by an invested amount of money (or principal) is reinvested so that it, too, earns interest.

40. Exponential Functions (7 of 9)
Example – Population Growth
The population of a town of 10,000 grows at the rate of 2% per year. Find the population three years from now.

41. Exponential Functions (8 of 9)
Example – Population Growth

42. Exponential Functions (9 of 9)
Example – Radioactive Decay

43. Logarithmic Functions (1 of 5)
Each exponential function has an inverse. These functions, inverse to the exponential functions, are called logarithmic functions.

44. Logarithmic Functions (2 of 5)
Example – Converting from Exponential to Logarithmic Form
Exponential Form
Logarithmic Form
a. Since
52 = 25
it follows that
log5 25 = 2
b. Since
34 = 81
log3 81 = 4
c. Since
100 = 1
log10 1 = 0

45. Logarithmic Functions (3 of 5)
Example – Graph of a Logarithmic Function with b > 1

46. Logarithmic Functions (4 of 5)
Example – Finding Logarithms

47. Logarithmic Functions (5 of 5)
Example 7 – Finding Half-Life

48. Properties of Logarithms (1 of 8)
The logarithm function has many important properties. For example:

49. Properties of Logarithms (2 of 8)
Example 1 – Finding Logarithms by Using a Table
Table 4.4 Common Logarithms x
log x 2
0.3010 7
0.8451 3
0.4771 8
0.9031 4
0.6021 9
0.9542 5
0.6990 10
1.0000 6
0.7782 e
0.4343

50. Properties of Logarithms (3 of 8)
Example 1 – Continued

51. Properties of Logarithms (4 of 8)
Example – Writing Logarithms in Terms of Simpler Logarithms

52. Properties of Logarithms (5 of 8)
Example – Continued

53. Properties of Logarithms (6 of 8)
Example – Simplifying Logarithmic Expressions

54. Properties of Logarithms (7 of 8)
Example – Continued

55. Properties of Logarithms (8 of 8)
Example – Evaluating a Logarithm Base 5

56. Logarithmic and Exponential Equations (1 of 6)
A logarithmic equation involves the logarithm of an expression containing an unknown.
An exponential equation has the unknown appearing in an exponent.

57. Logarithmic and Exponential Equations (2 of 6)
Example 1 – Oxygen Composition

58. Logarithmic and Exponential Equations (3 of 6)
Example 1 – Continued

59. Logarithmic and Exponential Equations (4 of 6)
Example – Using Logarithms to Solve an Exponential Equation

60. Logarithmic and Exponential Equations (5 of 6)
Example – Predator-Prey Relation

61. Logarithmic and Exponential Equations (6 of 6)
Example – Continued