1. MTH108 Business Math I Lecture 4

2. Chapter 2 Linear Equations

3. 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

4. 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

5. 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.

6. 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 form ax + by = c (2.1) where a, b and c are constants and a and b cannot both equal zero.

7. Examples Equation abc Variables 2 x +5 y =-525-5 x and y - u + v /2=01/20 u and v x /3=251/3025 x and y 2 s -4 t =-1/22-4-1/2 s and t

8. Examples Equation abc VariablesLinear/No n-linear 2 x +3 xy =-52?-5 x and y Non-linear -√ u + v /2=01/20 u and v Non-linear x+y 2 =251?25 x and y Non-linear 2 s -4/ t =-1/22-?-1/2 s and t Non-linear 2 x =(5 x -2 y )/4 +102,5210 x and y Linear

9. Verifying

10. 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

11. Examples 2x + 4y = 16 1)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

12. 3)Production possibilities

13. Production possibilities (contd.)

16. Linear equation with n variables Definition A linear equation involving n variables x 1, x 2,..., x n has the general form a 1 x 1 + a 2 x 2 +... + a n x n = b (2.2) where a 1, a 2,..., a n and b are constants not all a 1, a 2,..., a n equal zero.

17. Examples

18. 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={ ( )| a 1 x 1 + a 2 x 2 +... + a n x n = b } As in the case of two variables, there are infinitely many values in the solution set.

19. Example

20. Example (contd.)

22. 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

23. Examples 1)2 x +4 y = 16

24. 2)4 x -7 y =0 Any two variable linear equation having the form graphs a straight line which passes through the origin.

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

26. Examples

28. 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

29. 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