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Chapter 7: Systems of Equations and Inequalities; Matrices

Chapter 7: Systems of Equations and Inequalities; Matrices. 7.1 Systems of Equations 7.2 Solution of Linear Systems in Three Variables 7.3 Solution of Linear Systems by Row Transformations 7.4 Matrix Properties and Operations 7.5 Determinants and Cramer’s Rule

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Chapter 7: Systems of Equations and Inequalities; Matrices

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  1. Chapter 7: Systems of Equations and Inequalities; Matrices 7.1 Systems of Equations 7.2 Solution of Linear Systems in Three Variables 7.3 Solution of Linear Systems by Row Transformations 7.4 Matrix Properties and Operations 7.5 Determinants and Cramer’s Rule 7.6 Solution of Linear Systems by Matrix Inverses 7.7 Systems of Inequalities and Linear Programming 7.8 Partial Fractions

  2. 7.1 Systems of Equations • A linear equation in n unknowns has the form where the variables are of first-degree. • A set of equations is called a system of equations. • The solutions of a system of equations must satisfy every equation in the system. • If all equations in a system are linear, the system is a system of linear equations, or a linear system.

  3. 7.1 Linear System in Two Variables • Three possible solutions to a linear system in two variables: • One solution: coordinates of a point, • No solutions: inconsistent case, • Infinitely many solutions: dependent case.

  4. 7.1 Substitution Method (1) (2) Example Solve the system. Solution Solve (2) for y. Substitute y = x + 3 in (1). Solve for x. Substitute x = 1 in y = x + 3. Solution set: {(1, 4)}

  5. 7.1 Solving a Linear System in Two Variables Graphically (1) (2) Example Solve the system graphically. Solution Solve (1) and (2) fory.

  6. 7.1 Elimination Method (1) (2) Example Solve the system. Solution To eliminate x, multiply (1) by –2 and (2) by 3 and add the resulting equations. (3) (4)

  7. 7.1 Elimination Method Substitute 2 for y in (1) or (2). The solution set is {(3, 2)}. • Check the solution set by substituting 3 in for x and 2 in for y in both of the original equations.

  8. 7.1 Solving an Inconsistent System (1) (2) Example Solve the system. Solution Eliminate x by multiplying (1) by 2 and adding the result to (2). Solution set is . Inconsistent System

  9. 7.1 Solving a System with Dependent Equations (1) (2) Example Solve the system. Solution Eliminate x by multiplying (1) by 2 and adding the result to (2). Each equation is a solution of the other. Choose either equation and solve for x. The solution set is e.g. y = –2:

  10. 7.1 Solving a Nonlinear System of Equations (1) (2) Example Solve the system. Solution Choose the simpler equation, (2), and solve for y since x is squared in (1). Substitute for y into (1) .

  11. 7.1 Solving a Nonlinear System of Equations Substitute these values for x into (3). The solution set is

  12. 7.1 Solving a Nonlinear System Graphically Example Solve the system. Solution (1) yields Y1 = 2x; (2) yields Y2 = |x + 2|. The solution set is {(2, 4), (–2.22, .22), (–1.69, .31)}.

  13. 7.1 Applications of Systems • Solving Application Problems • Read the problem carefully and assign a variable to represent each quantity. • Write a system of equations involving these variables. • Solve the system of equations and determine the solution. • Look back and check your solution. Does it seem reasonable?

  14. 7.1 Applications of Systems Example In a recent year, the national average spent on two varsity athletes, one female and one male, was $6050 for Division I-A schools. However, average expenditures for a male athlete exceeded those for a female athlete by $3900. Determine how much was spent per varsity athlete for each gender.

  15. 7.1 Applications of Systems Solution Let x = average expenditures per male y = average expenditures per female Average spent on one male and onefemale Average Expenditure per male: $8000, and per female: from (2) y = 8000 – 3900 = $4100.

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