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Transcript
Chapter 2 Systems of Linear Equations and Matrices • 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 2.1 Systems of Linear Equations: An Introduction • Recall that a system of two linear equations in two variables may be written in the general form ax by h cx dy k 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 L1 and L2, represented by the first and second equations of the system. Systems of Equations • Given the two straight lines L1 and L2, one and only one of the following may occur: 1. L1 and L2 intersect at exactly one point. y L1 y1 Unique solution (x1, y1) (x1, y1) x1 x L2 Systems of Equations • Given the two straight lines L1 and L2, one and only one of the following may occur: 2. L1 and L2 are coincident. y L1, L2 Infinitely many solutions x Systems of Equations • Given the two straight lines L1 and L2, one and only one of the following may occur: 3. L1 and L2 are parallel. y L1 L2 No solution x Example: a system of equations with exactly one solution • Consider the system 3x y 2 x y 10 • Solving the first equation for y in terms of x, we obtain y 3x 2 • Substituting this expression for y into the second equation yields x (3x 2) 10 4 x 2 10 4 x 12 x3 Example: a system of equations with exactly one solution • Finally, substituting this value of x into the expression for y obtained earlier gives y 3x 2 3(3) 2 7 • Therefore, the unique solution of the system is given by x = 3 and y = 7. 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 12 3x y 2 10 8 (3, 7) 6 4 x y 10 2 –2 2 4 6 8 10 12 x Example: a system of equations with infinitely many solutions • • Consider the system 2 x y 4 4x 2 y 8 Solving the first equation for y in terms of x, we obtain y 2x 4 • Substituting this expression for y into the second equation yields 4 x 2(2 x 4) 8 4x 4x 8 8 00 • • which is a true statement. This result follows from the fact that the second equation is equivalent to the first. 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. 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 8 6 2 x y 4 4x 2 y 8 4 2 -6 -4 -2 -2 -4 –6 2 4 6 x Example: a system of equations that has no solution • • Consider the system 2 x y 4 4 x 2y 3 Solving the first equation for y in terms of x, we obtain y 2x 4 • Substituting this expression for y into the second equation yields 4 x 2(2 x 4) 3 4x 4x 8 3 0 5 • • which is clearly impossible. Thus, there is no solution to the system of equations. 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: y 3 4x 2 y 3 2 x y 4 2 1 -2 -1 -1 -2 -3 –4 1 2 3 4 x A System of 3 Linear Equations a1 x b1 y c1 z d1 a2 x b2 y c2 z d 2 a3 x b3 y c3 z d3 can have Infinitely many solutions (line) No solution (parallel) Linear Equations with n Variables • A linear equation in n variables x1 , x2 ,..., xn is one of the form a1 x1 a2 x2 ... an xn c where a1 , a2 ,..., an not all zero and c are constant. Ex. 2x1 3x2 4 x3 5 x4 10 x5 12 is a linear equation in the five variables, x1 , x2 , x3 , x4 , and x5 . 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.) 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)