1. MTH108 Business Math I
Lecture 6

2. Systems of Linear Equations
Chapter 3
Systems of Linear Equations

3. 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

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

5. 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

6. 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

7. 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.

8. 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.

9. 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

10. 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

11. 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

12. Graphical solutions

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

14. 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

15. Examples 1)

16. Examples (contd.) 2)

17. Examples (contd.) 3)

18. (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

19. Example

20. Example (contd.)

21. Example (contd.)
Section 3.1

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

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