MTH108 Business Math I

Lecture 4

Chapter 2

Linear Equations

Objectives

• Provide a thorough understanding of the algebraic and graphical characteristics of linear equations

• Provide the tools which allow one to determine the equation which represents a linear relationship

• Illustrate some applications

Today’s Topics

• Importance of linear mathematics• Characteristics of linear equations• Solution set• Linear equations with n variables; solution set

and examples• Graphing linear equation of two variables• Solution set, intercepts

Linear Mathematics

Study of linear mathematics is important in many ways.

• Many real world problems can be mathematically represented in a linear relationship

• Analysis of linear relationships is easier than non-linear ones

• Methods of analysing non-linear relationships are mostly similar to, or extensions of linear ones

Thus understanding of linear mathematics is important to study non-linear mathematics.

Characteristics of Linear Equations

Recall that a variable is a symbol that can be replaced by any one of a set of different numbers. e.g. 10- x.

Definition A linear equation involving two variables x and y has

the standard formax + by= c (2.1)

where a, b and c are constants and a and b cannot both equal zero.

Examples

Equation a b c Variables

2x+5y=-5 2 5 -5 x and y

-u+v/2=0 -1 1/2 0 u and v

x/3=25 1/3 0 25 x and y

2s-4t=-1/2 2 -4 -1/2 s and t

ExamplesEquation a b c Variables Linear/

Non-linear2x+3xy=-5 2 ? -5 x and y Non-linear

-√u+v/2=0 -1 1/2 0 u and v Non-linear

x+y2=25 1 ? 25 x and y Non-linear

2s-4/t=-1/2 2 -? -1/2 s and t Non-linear

2x=(5x-2y)/4 +10 2,5 2 10 x and y Linear

Verifying

Solution set of an equation

Given a linear equation ax + by= c, the solution set for the equation (2.1) is the set of all ordered pairs (x, y) which satisfy the equation.

S={(x,y)|ax+by=c} For any linear equation, S consists of an infinite number

of elements.Method:• Assume a value of one variable• Subtitute this into the equation• Solve for the other variable

Examples

2x + 4y= 161) Determine the pair of values which satisfy the

equation when x=-2

2) Determine the pair of values which satisfy the equation when y=0

3) Production possibilities

Production possibilities (contd.)

Production possibilities (contd.)

Production possibilities (contd.)

Linear equation with n variables

DefinitionA linear equation involving n variables x1, x2, . . . , xn

has the general form

a1 x1 + a2 x2+ . . . + an xn = b (2.2)

where a1, a2, . . . , an and b are constants not all a1, a2, . . . ,

an equal zero.

Examples

The solution set of a linear equation with n variables as defined in (2.2) is the n-tuple ( )

satisfying (2.2). The set S will be

S={ ( )| a1 x1 + a2 x2+ . . . + an xn = b }

As in the case of two variables, there are infinitely many values in the solution set.

Example

Example (contd.)

Example (contd.)

Graphing two variable equations

A linear equation involving two variables graphs as a straight line in two dimensions.

Method:• Set one variable equal to zero• Solve for the value of other variable• Set second variable equal to zero• Solve for the value of first variable• The ordered pairs (0, y) and (x, 0) lie on the line

Examples

1) 2x+4y = 16

2) 4x-7y=0

• Any two variable linear equation having the form graphs a straight line which passes through the origin.

Interceptsx-interceptThe x-intercept of an equation is the point where the

graph of the equation crosses the x-axis,i.e. y=0Y-interceptThe y-intercept of an equation is the point where the

graph of the equation crosses the y-axis,i.e. x=0• Equations of the form x=k has no y-intercept• Equations of the form y=k has no x-intercept

Examples

Examples

Summary

• Importance of linear mathematics• Characteristics of linear equations• Linear equations with examples• Solution set of an equation• Linear equation with n variables• Graphing two variable equations• Intercepts • Section 2.1 follow-up exercises• Section 2.2 Q.1-37

Next lecture

• Slope of an equation• Slope-intercept form• One-point form• Two-point form• Parallel and perpendicular lines• Linear equations involving more than two variables• Some applications