1. Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e
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2. Systems of Linear Equations and Matrices
Chapter 6
Systems of Linear Equations and Matrices
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3. Systems of Two Linear Equations in Two Variables
Section 6.1
Systems of Two Linear Equations in Two Variables
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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.
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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.
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8. Larger Systems of Linear Equations
Section 6.2
Larger Systems of Linear Equations
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19. 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.
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24. Applications of Systems of Linear Equations
Section 6.3
Applications of Systems of Linear Equations
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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 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,
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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:
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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
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32. Basic Matrix Operations
Section 6.4
Basic Matrix Operations
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34. 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
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43. Matrix Products and Inverses
Section 6.5
Matrix Products and Inverses
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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
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52. Applications of Matrices
Section 6.6
Applications of Matrices
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54. 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.
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