MTH108 Business Math I Lecture 6

Systems of Linear Equations Chapter 3 Systems of Linear Equations

Review Chapter 1 Chapter 2 Solving equations and inequalities in one variable Absolute value Rectangular coordinate systems Chapter 2 Characteristics of linear equations Graphical characteristics Slope-Intercept form

Review (contd.) Determining the equation of a straight line Linear equations involving more than two variables

Objectives Provide an understanding of the nature of systems of equations and their graphical representation Provide an understanding of different solution set possibilities and their graphical interpretation Present procedures for determining solution sets for system of equations Some applications

Today’s topics Two variable systems of equations Solution sets Graphical analysis The elimination procedure Systems of more than two equations with two variables only

System of Equations Given a set of equations, we will be concerned with the process used to determine whether there are values of variables which jointly satisfy a set of equations. e.g. whether there are quantities of the four items which satisfy the attributes of weight capacity, volume capacity, budget and water requirements.

System of Equations A system of Equations is a set consisting of more than one equation. we use dimensions to characterize a system of equations. If m denotes the no. of equations and n denotes the no. of variables, then the system is called an “m by n system”. In solving systems of equations, we are interested in identifying values of the variables that satisfy all equations in the system simultaneously.

Solution set Example We may wish to identify any values of x and y which satisfy both equations at the same time, i.e. Three possibilities of S: Null set, finite set, infinite set

Graphical Analysis Linear equation with two variables----straight line System of linear equations with two variables----two straight lines To determine whether the two lines have any point in common

Slope-Intercept relationships Given a (2 × 2) system of linear equations (in slope-intercept form), where and represent the two slopes and and denote the two y-intercepts. unique solution, if No solution, if Infinite solutions, if

Graphical solutions

Graphical solutions Verification: Limitations: good for two variable system of equations Not good for non-integer values Algebraic solution is preferred

The Elimination Procedure Given a (2 × 2) system of equations, Eliminate of the variable by multiplying or adding the two equations Solve the remaining equation in terms of remaining variable Substitute back into one of the given equation to find the value of the eliminated variable

Examples 1)

Examples (contd.) 2)

Examples (contd.) 3)

(m × 2) systems, m >2 When there are more than two equations involving only two variables, We solve two equations first And then put the values in the rest of the equations

Example

Example (contd.)

Example (contd.) Section 3.1

Review Two variable systems of equations Solution sets Graphical analysis The elimination procedure Systems of more than two equations with two variables only

Next lecture Gaussian elimination method N-variable systems n >3