Section 7.1 |
Section 7.3 |
Section 7.5 |

Section 7.2 |
Section 7.4 |

Students will be able to solve systems of equations graphically and algebraically.

**Key Ideas**

Demand curve | Solution by substitution |

Equilibrium point | Solution of a system |

Equilibrium price | Supply curve |

Solution by elimination | System of equations |

**Teaching Tips**

Many students will have already seen the substitution and elimination methods for solving systems of linear equations, but a light review does not hurt. Be sure to emphasize the graphical interpretation of finding intersection points of lines. Not only does this clarify the phenomena of empty solutions and infinite solutions, but it serves as an easy bridge to the graphical solution of non-linear systems. Algebraic solutions of non-linear systems can be quite difficult in general, so our non-linear, algebraic examples are deliberately chosen to be simple.

**Technology Tips**

Calculators that will solve linear systems can obscure the algebraic theory that this section is intended to highlight, so explain to students that we are less interested (for now) in knowing they can get the answers than we are in knowing that they *understand* these pencil-and-paper methods. Assure them that we will all get technological in Section 7.3.

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Students will be able to find sums, differences, scalar multiples, and products of matrices. They will be able to find the inverses of matrices by hand and the inverses of larger square matrices by using a calculator.

**Key Ideas**

Additive inverse | Non-singular matrix |

Cofactor | Order of a matrix |

Column | Row |

Determinant | Scalar |

Element (entry) of a matrix | Singular matrix |

Inverse of a matrix | Square matrix |

Matrix | Transpose of a matrix |

Minor | Zero matrix |

**Teaching Tips**

Teachers might well decide to introduce this material much earlier in the course, since it is not dependent on anything in the first six chapters. Also, there are advantages to having matrices available for examples in other sections. For example, it is an interesting bit of trigonometry to prove that

Be sure that students understand the application of matrix multiplication in Example 5. (There are several exercises that test this understanding.) Computer spreadsheets do matrix multiplications in the blink of an eye, but it is still up to computer users to set up the multiplications properly.

The fact that matrix multiplication is *not commutative* is a significant algebraic discovery for students and should not be passed over lightly.

The general determinant algorithm of expansion by cofactors, given on page 559, is shown in this section for the sake of completeness. It is not recommended that students find determinants by hand for matrices larger than , unless they know enough about row-reduction to make the task reasonable. (For example, students who compete in mathematics contests might be challenged to find larger determinants, but the secret is usually to reduce them first.)

There is a fairly well-known trick for computing determinants, but teaching it to students might do more harm than good since it does not generalize at all to matrices of other orders. If the object is to understand determinants, teach cofactors. If the object is to find determinants, use a calculator. In this course, the object is usually to find determinants (or, more often, to find inverse matrices).

**Technology Tips**

Feel free to use the calculator quite freely in this section. Matrices were invented to take some of the tedium out of computations, and if the calculator can make things even less tedious, then they are right in step with the program.

The section has quite a few technology tips in it already, but here is one more. Not many calculator users realize that they can build matrices on the home screen by using square brackets. For example, [[1,2,3][4,5,6][7,8,9]] is read as a matrix. The following screen shows how to type it on the home screen and store it as matrix [A]:

Students will be able to solve systems of linear equations using Gaussian elimination, the reduced row echelon form of a matrix, or matrix inversion.

**Key Ideas**

Augmented matrix | Invertible square linear system |

Coefficient matrix | Reduced row echelon form |

Equivalent linear systems | Row echelon form |

Gaussian elimination | Triangular form |

**Teaching Tips**

The material in this section is not necessary for first-year calculus, but it explains the mathematics behind two time-saving calculator approaches to solving systems of linear equations. Since students will find the technique of Examples 7, 8, and 9 to be the easiest, teachers who are pressed for time might want to consider omitting the other examples. Note, however, that the solution only works on *invertible square linear systems.* Example 3 (no solution) and Examples 5 and 6 (infinitely many solutions) are best solved by Gaussian elimination or by interpreting the reduced row echelon form of the augmented matrix. Note that the calculator can be used to find reduced row echelon form.

**Technology Tips**

The solving of simultaneous linear equations is important for many real-world problems in elementary algebra, and most of them are modeled by invertible square linear systems. Since the modeling in this case is more important than the manipulations required to solve the systems, the day is not far off when Algebra I students will be solving such problems using (or possibly some user-friendly menu item) on their graphing calculators. The slower and riskier "pencil-and-paper" methods should therefore be studied for their historical and pedagogical value, not because they will actually be used.

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Students will be able to decompose certain rational expressions into partial fractions.

**Key Ideas**

Partial fractions

Partial fraction decomposition

**Teaching Tips**

This little nugget of algebra is useful for finding antiderivatives of rational expressions in first-year calculus, although many calculus teachers skip those antiderivatives precisely because students do not know how to find the partial fractions. Realistically, it is not necessary for students to learn this process well. Those who have calculus teachers who like the topic will have it explained to them again, and those who have calculus teachers who skip the topic might never see it at all.

The AB Calculus course description for AP Calculus does *not* include partial fractions. The topic *is* in the BC course description, but students only need to be able to decompose rational expressions with unrepeated linear factors in the denominator (as in Example 1).

**Technology Tips**

Graphing calculators with computer algebra systems (CAS) will decompose rational expressions into partial fractions. Indeed, they will find the solutions to the differential equations that partial fractions are needed to solve. These two facts suggest that it is only a matter of time before this topic disappears from calculus and precalculus courses entirely. Meanwhile, if you want your students to learn the algebraic manipulations required to find partial fractions, be sure to restrict the use of CAS calculators.

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Students will be able to solve linear programming problems and systems of inequalities using graphical methods.

**Key Ideas**

Constraints | Objective function |

Half-plane | System of inequalities |

Linear programming problem | Vertex (corner) points of a feasible region |

**Teaching Tips**

Linear inequalities are covered in this section so that linear programming can be included in the textbook. (Linear programming is a syllabus item on some state requirement lists.) As far as preparation for calculus is concerned, this section is an interesting detour.

Notice the shading of the region in Figure 7.32. Shading *only* the region that satisfies all the constraints is vastly preferable to shading the constraints one at a time and searching the cross-hatched diagram for the intersection of all the shadings.

**Technology Tips**

Shown below is the way to select the shading "styles" necessary to shade the inequalities in Example 7. The result, as you can see, is too muddy to be very useful.

A clever way to get a readable graph is to shade everything in the opposite direction! The feasible region that satisfies all the constraints then shows up in white, as shown below.