Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All right reserved.

Systems of Linear Equations and Matrices Chapter 6 Systems of Linear Equations and Matrices Copyright ©2015 Pearson Education, Inc. All right reserved.

Systems of Two Linear Equations in Two Variables Section 6.1 Systems of Two Linear Equations in Two Variables Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Solve the system Solution: Multiply the first equation by 2 and the second equation by −3 to get The multipliers 2 and −3 were chosen so that the coefficients of x in the two equations would be negatives of each other. Any solution of both of these equations must also be a solution of their sum: To find the corresponding value of x, substitute 2 for y in either of the original equations. Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Solve the system Solution: We choose the first equation: Therefore, the solution of the system is The graphs of both equations of the system are shown below. They intersect at the point the solution of the system. Copyright ©2015 Pearson Education, Inc. All right reserved.

Larger Systems of Linear Equations Section 6.2 Larger Systems of Linear Equations Copyright ©2015 Pearson Education, Inc. All right reserved.

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Use matrices to solve the system Example: Use matrices to solve the system Solution: Write the augmented matrix and perform row operations to obtain a first column whose entries (from top to bottom) are 1, 0, 0: Stop! The second row of the matrix denotes the equation Since the left side of this equation is always zero and the right side is 2, it has no solution. Therefore, the original system has no solution. Copyright ©2015 Pearson Education, Inc. All right reserved.

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Applications of Systems of Linear Equations Section 6.3 Applications of Systems of Linear Equations Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Example: The table shows Census Bureau projections for the population of the United States (in millions). Use the given data to construct a quadratic function that give the U.S. population (in millions) in year x. Solution: Let correspond to the year 2000. Then the table represents data points (22, 334), (40, 380), and (50, 400). We must find a function of the form whose graph contains these three points. If (20, 334) is to be on the graph, we must have that is, Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Example: The table shows Census Bureau projections for the population of the United States (in millions). Use the given data to construct a quadratic function that give the U.S. population (in millions) in year x. Solution: The other two points lead to these equations: Now work by hand or use technology to solve the following system: Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Example: The table shows Census Bureau projections for the population of the United States (in millions). Use the given data to construct a quadratic function that give the U.S. population (in millions) in year x. Solution: The reduced row echelon form of the augmented matrix shown in the figure below shows that the solution is So the function is Copyright ©2015 Pearson Education, Inc. All right reserved.

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Basic Matrix Operations Section 6.4 Basic Matrix Operations Copyright ©2015 Pearson Education, Inc. All right reserved.

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Find each sum if possible. Example: Find each sum if possible. (a) Solution: Because the matrices are the same size, they can be added by finding the sum of each corresponding element. (b) Solution: The matrices are of different sizes, so it is not possible to find the sum Copyright ©2015 Pearson Education, Inc. All right reserved.

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Matrix Products and Inverses Section 6.5 Matrix Products and Inverses Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Given matrices A and B as follows, determine whether B is the inverse of A: Solution: B is the inverse of A if so we find those products. Therefore, B is the inverse of A; that is, (It is also true that A is the inverse of B, or Copyright ©2015 Pearson Education, Inc. All right reserved.

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Applications of Matrices Section 6.6 Applications of Matrices Copyright ©2015 Pearson Education, Inc. All right reserved.

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Use the inverse of the coefficient matrix to solve the system Example: Use the inverse of the coefficient matrix to solve the system Solution: The coefficient matrix is A graphing calculator will indicate that does not exist. If we carry out the row operations, we see why: The next step cannot be performed because of the zero in the second row, second column. Therefore, the inverse does not exist, and the system has no solutions. Copyright ©2015 Pearson Education, Inc. All right reserved.

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