Copyright © 2009 Pearson Education, Inc. CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations in Three Variables 9.3 Matrices and Systems of Equations 9.4 Matrix Operations 9.5 Inverses of Matrices 9.6 Determinants and Cramer’s Rule 9.7 Systems of Inequalities and Linear Programming 9.8 Partial Fractions

Copyright © 2009 Pearson Education, Inc. 8.1 Systems of Equations in Two Variables  Solve a system of two linear equations in two variables by graphing.  Solve a system of two linear equations in two variables using the substitution and the elimination methods.  Use systems of two linear equations to solve applied problems.

Slide Copyright © 2009 Pearson Education, Inc. Systems of Equations A system of equations is composed of two or more equations considered simultaneously. Example: 5x  y = 5 4x  y = 3 This is a system of two linear equations in two variables. The solution set of this system consists of all ordered pairs that make both equations true. The ordered pair (2, 5) is a solution of this system.

Slide Copyright © 2009 Pearson Education, Inc. Solving Systems of Equations Graphically When we graph a system of linear equations, each point at which the graphs intersect is a solution of both equations and therefore a solution of the system of equations.

Slide Copyright © 2009 Pearson Education, Inc. Solving Systems of Equations Graphically Let’s solve the previous system graphically. 5x  y = 5 4x  y = 3 Solution: We see that the graph intersects at the single point (2, 5), so this is the solution of the system of equations.

Slide Copyright © 2009 Pearson Education, Inc. Systems of Equations If a system of equations has at least one solution, it is consistent. If the system has no solutions, it is inconsistent. If a system of two linear equations in two variables has an infinite number of solutions, the equations are dependent. Otherwise, they are independent.

Slide Copyright © 2009 Pearson Education, Inc. Illustration of Graphs Graphs of linear equations may be related to each other in one of three ways.

Slide Copyright © 2009 Pearson Education, Inc. Substitution Method The substitution method is a technique that gives accurate results when solving systems of equations. It is most often used when a variable is alone on one side of an equation or when it is easy to solve for a variable. One equation is used to express one variable in terms of the other, then it is substituted in the other equation.

Slide Copyright © 2009 Pearson Education, Inc. Example Use substitution to solve the system 5x  y = 5, 4x  y = 3. Solution Solve the first equation for y: y = 5x  5 Then we substitute 5x  5 for y in the second equation to give an equation in one variable. 4x  (5x  5) = 3 4x  5x + 5 = 3 x = 2

Slide Copyright © 2009 Pearson Education, Inc. Solution continued Now we use back-substitution and substitute 2 for x in either original equation. 4x  y = 3 4(2)  y = 3 8  y = 3 y = 5 We find the solution to the system of equations to be (2, 5), once again.

Slide Copyright © 2009 Pearson Education, Inc. Elimination Method Using the elimination method, we eliminate one variable by adding the two equations. If the coefficients of a variable are opposites, that variable can be eliminated by simply adding the original equations. If the coefficients are not opposites, it is necessary to multiply one or both equations by suitable constants, before we add.

Slide Copyright © 2009 Pearson Education, Inc. Example Solve the system using the elimination method. 6x + 2y = 4 10x + 7y =  8 Solution If we multiply the first equation by 5 and the second equation by  3, we will be able to eliminate the x variable. 30x + 10y = 20 Substituting: 6x + 2y = 4  30x  21y = 24 6x + 2(  4) = 4  11y = 44 6x  8 = 4 y =  4 6x = 12 The solution is (2,  4). x = 2

Slide Copyright © 2009 Pearson Education, Inc. Another Example Solve the system. x  3y =  9 (1) 2x  6y = 3 (2) Solution:  2x + 6y = 18 Mult. (1) by  2 2x  6y = 3 0 = 21 There are no values of x and y in which 0 = 21. So this system has no solution. The graphs of the equations are of parallel lines.

Slide Copyright © 2009 Pearson Education, Inc. Another Example Solve the system. 9x + 6y = 48 (1) 3x + 2y = 16 (2) Solution: 9x + 6y = 48  9x  6y =  48 Mult. (2) by  3 0 = 0 When we obtain the equation 0 = 0, we know the equations are dependent. There are infinitely many solutions. The graphs of the equations are identical.

Slide Copyright © 2009 Pearson Education, Inc. Application Ethan and Ian are twins. They have decided to save all of the money they earn, at their part-time jobs, to buy a car to share at college. One week, Ethan worked 8 hours and Ian worked 14 hours. Together they saved $256. The next week, Ethan worked 12 hours and Ian worked 16 hours and they earned $324. How much does each twin make per hour?

Slide Copyright © 2009 Pearson Education, Inc. Solution Letting E represent Ethan and I represent Ian, the following system can be obtained. 8E + 14I = 256 First week 12E + 16I = 324 Second week Mult by 12 96E + 168I = 3072 Mult by  8  96E  128I =  I = 480 I = 12

Slide Copyright © 2009 Pearson Education, Inc. Solution Solve for E. 8E + 14(12) = 256 8E = 88 E = 11 Ian makes $12 per hour while Ethan makes $11 per hour.