1. Copyright © 2008 Pearson Education Canada4-1 Chapter 4 Linear Systems Contemporary Business Mathematics with Canadian Applications Eighth Edition S. A. Hummelbrunner/K. Suzanne Coombs PowerPoint: D. Johnston

2. Copyright © 2008 Pearson Education Canada4-2 Objectives After completing chapter four the student will be able to: Use of Linear System in deciding the combination of production levels to use of labour, materials, and produce highest level of profit. Use of Linear System to decide how to best allocate business resources. Use the method of elimination to solve a linear system of two simultaneous equations in two variables. Graph linear equations. Graph linear systems of two equations with two variables. Solve problems using linear systems.

3. Copyright © 2008 Pearson Education Canada4-3 Algebraic Elimination Method

4. Copyright © 2008 Pearson Education Canada4-4 Algebraic Elimination Method (continued)

5. Copyright © 2008 Pearson Education Canada4-5 Unequal Coefficients

6. SOLVE Copyright © 2008 Pearson Education Canada4-6 1. y = 3x + 12 x = –y 2. 6x + 3y = 24 2x + 9y = –8

7. Copyright © 2008 Pearson Education Canada4-7 Graphing Linear Equations in a system of rectangular coordinates

8. Copyright © 2008 Pearson Education Canada4-8 Locating a Point

9. Constructing a table of values To graph a linear equation a suitable set of points may be obtained by constructing a table of values. Substitute random value for X or Y and computer the value of the second variable. List ordered pairs (X,Y) in a table of values. Copyright © 2008 Pearson Education Canada4-9

10. Copyright © 2008 Pearson Education Canada4-10 Constructing a Table for x + y = 4 or y = 4 - x

11. SOLVE Copyright © 2008 Pearson Education Canada4-11 Construct a table of values for X = Y – 2, for values of Y from -3 to +5

12. Copyright © 2008 Pearson Education Canada4-12 Graphing an Equation x + y = 4

13. Copyright © 2008 Pearson Education Canada4-13 Special Cases Lines Parallel to X-axis Note the graphs for y = 3 and y = -3.

14. Copyright © 2008 Pearson Education Canada4-14 Special Cases Lines Parallel to Y-axis Note the graphs of x =3 and x = -3.

15. Copyright © 2008 Pearson Education Canada4-15 Slope-y-intercept Form of a Linear Equation

16. Copyright © 2008 Pearson Education Canada4-16 Writing an Equation in Slope-y-intercept Form

17. Copyright © 2008 Pearson Education Canada4-17 Special Cases for the Slope

18. Copyright © 2008 Pearson Education Canada4-18 Different Cases for the Slope

19. SOLVE Copyright © 2008 Pearson Education Canada4-19 Page 162 Exercise 4.2 Write the coordinates A – 1 Plot the point in system of rectangular axes A – 2 Page 163 Exercise 4.2 Graph the equations B – 1 B – 2

20. Copyright © 2008 Pearson Education Canada4-20 Graph a System of Linear Equations x = y x-2y +2000 = 0

21. Copyright © 2008 Pearson Education Canada4-21 Problem Solving

22. Copyright © 2008 Pearson Education Canada4-22 Problem Solving (continued)

23. SOLVE following Linear Systems graphically Copyright © 2008 Pearson Education Canada4-23 1.X + Y = 4 and X – Y = - 4 2.5X – 2Y = 20 and Y = 5

24. Copyright © 2008 Pearson Education Canada4-24 Summary Linear systems of two simultaneous equations in two variables are useful for modelling business applications. Such systems can be solved by graphing the two linear equations and finding the point of intersection. Such systems can be solved algebraically by using the method of elimination.