1. Systems of Linear Equations: An Introduction Unique Solutions Underdetermined and Overdetermined Systems Multiplication of Matrices The Inverse of a Square Matrix Leontief Input-Output Model Chapter 2 Systems of Linear Equations and Matrices

2. Recall that a system of two linear equations in two variables may be written in the general form where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero. Recall that the graph of each equation in the system is a straight line in the plane, so that geometrically, the solution to the system is the point(s) of intersection of the two straight lines L 1 and L 2, represented by the first and second equations of the system. 2.1 Systems of Linear Equations: An Introduction

3. Systems of Equations Given the two straight lines L 1 and L 2, one and only one of the following may occur: 1. L 1 and L 2 intersect at exactly one point. y x L1L1L1L1 L2L2L2L2 Unique solution (x 1, y 1 ) x1x1x1x1 y1y1y1y1

4. Systems of Equations Given the two straight lines L 1 and L 2, one and only one of the following may occur: 2. L 1 and L 2 are coincident. y x L 1, L 2 Infinitely many solutions

5. Systems of Equations Given the two straight lines L 1 and L 2, one and only one of the following may occur: 3. L 1 and L 2 are parallel. y x L1L1L1L1 L2L2L2L2 No solution

6. Example: a system of equations with exactly one solution Consider the system Solving the first equation for y in terms of x, we obtain Substituting this expression for y into the second equation yields

7. Example: a system of equations with exactly one solution Finally, substituting this value of x into the expression for y obtained earlier gives Therefore, the unique solution of the system is given by x = 3 and y = 7.

8. 24681012 24681012 12108642–2 Example: a system of equations with exactly one solution Geometrically, the two lines represented by the two equations that make up the system intersect at the point (3, 7): y x (3, 7)

9. Example: a system of equations with infinitely many solutions Consider the system Solving the first equation for y in terms of x, we obtain Substituting this expression for y into the second equation yields which is a true statement. This result follows from the fact that the second equation is equivalent to the first.

10. Example: a system of equations with infinitely many solutions Thus, any order pair of numbers (x, y) satisfying the equation y = 2x – 4 constitutes a solution to the system. By assigning the value t to x, where t is any real number, we find that y = 2t – 4 and so the ordered pair (t, 2t –4) is a solution to the system. The variable t is called a parameter. For example: Setting t = 0, gives the point (0, –4) as a solution of the system. Setting t = 1, gives the point (1, -2) as another solution of the system.

11. 8642-2-4–6 -6 -4 - 2 246 Example: a system of equations with infinitely many solutions Since t represents any real number, there are infinitely many solutions of the system. Geometrically, the two equations in the system represent the same line, and all solutions of the system are points lying on the line: y x

12. Example: a system of equations that has no solution Consider the system Solving the first equation for y in terms of x, we obtain Substituting this expression for y into the second equation yields which is clearly impossible. Thus, there is no solution to the system of equations.

13. -2- 1 1234 Example: a system of equations that has no solution To interpret the situation geometrically, cast both equations in the slope-intercept form, obtaining y = 2x – 4 and y = 2x – 3/2 which shows that the lines are parallel. Graphically: 321-2-3–4 y x

14. A System of 3 Linear Equations can have Unique solution Infinitely many solutions (line) No solution (parallel)

15. Linear Equations with n Variables A linear equation in n variables is one of the form where and c are constant.

16. Example The total number of passengers riding a certain city bus during the morning shift is 1000. If the child’s fare is $0.50, the adult’s fare is $1.25, and the total revenue from the fares in the morning shift is $987.5, how many children and how many adults rode the bus during the morning shift? (Formulate but do not solve the problem.)

17. Solution Let x = the number of children who rode the bus during the morning shift Let y = the number of adults who rode the bus during the morning shift We have the system of linear equations: x + y = 1000 (total passengers) 0.50x + 1.25y = 987.5 (total revenue)