848 CHAPTER 11 Systems of Equations and Inequalities Linear programming problem (p. 841) Maximize (or minimize) a linear objective function, = + z Ax By, subject to certain conditions, or constraints, expressible as linear inequalities in x and y. A feasible point ( ) x y , is a point that satisfies the constraints (linear inequalities) of a linear programming problem. Location of the solution of a linear programming problem (p. 842) If a linear programming problem has a solution, it is located at a corner point of the graph of the feasible points. If a linear programming problem has multiple solutions, at least one of them is located at a corner point of the graph of the feasible points. In either case, the corresponding value of the objective function is unique. Objectives Section You should be able to . . . Example(s) Review Exercises 11.1 1 Solve systems of equations by substitution (p. 758) 4 1–7, 56, 59 2 Solve systems of equations by elimination (p. 758) 5, 6 1–7, 56, 59 3 Identify inconsistent systems of equations containing two variables (p. 760) 7 5, 54 4 Express the solution of a system of dependent equations containing two variables (p. 760) 8 7, 53 5 Solve systems of three equations containing three variables (p. 761) 9, 12 8–10, 55, 57, 60 6 Identify inconsistent systems of equations containing three variables (p. 763) 10 10 7 Express the solution of a system of dependent equations containing three variables (p. 763) 11 9 11.2 1 Write the augmented matrix of a system of linear equations (p. 770) 1 20–25 2 Write the system of equations from the augmented matrix (p. 771) 2 11, 12 3 Perform row operations on a matrix (p. 771) 3, 4 20–25 4 Solve a system of linear equations using matrices (p. 773) 5–10 20–25 11.3 1 Evaluate 2 by 2 determinants (p. 784) 1 26 2 Use Cramer’s Rule to solve a system of two equations containing two variables (p. 785) 2 29, 30 3 Evaluate 3 by 3 determinants (p. 787) 4 27, 28 4 Use Cramer’s Rule to solve a system of three equations containing three variables (p. 789) 5 31 5 Know properties of determinants (p. 791) 6–9 32, 33 11.4 1 Find the sum and difference of two matrices (p. 796) 3, 4 13 2 Find scalar multiples of a matrix (p. 798) 5 14 3 Find the product of two matrices (p. 799) 6–11 15, 16 4 Find the inverse of a matrix (p. 803) 12–14 17–19 5 Solve a system of linear equations using an inverse matrix (p. 807) 15 20–25 11.5 1 Decompose P Q where Q has only nonrepeated linear factors (p. 814) 2 34 2 Decompose P Q where Q has repeated linear factors (p. 815) 3, 4 35 3 Decompose P Q where Q has a nonrepeated irreducible quadratic factor (p. 817) 5 36, 38 4 Decompose P Q where Q has a repeated irreducible quadratic factor (p. 818) 6 37 11.6 1 Solve a system of nonlinear equations using substitution (p. 821) 1, 3 39–43 2 Solve a system of nonlinear equations using elimination (p. 822) 2, 4, 5 39–43 11.7 1 Graph an inequality by hand (p. 831) 2–4 44, 45 2 Graph an inequality using a graphing utility (p. 833) 5, 7 44, 45 3 Graph a system of inequalities (p. 834) 6–12 46–50, 58 11.8 1 Set up a linear programming problem (p. 840) 1 61 2 Solve a linear programming problem (p. 841) 2–4 51, 52, 61
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