Ch02+Systems+of+Equations

Chapter Two - Systems of Equations and inequalities
Chapter objectives, topics, and assignments:

Chapter Two highlights in Powerpoint:

And here they are as a pdf file:

2-1 Solving Systems of (Linear) Equations

 * Method 1: Graph the two equations; the solution is where the lines cross.
 * If the lines cross, then the solution is given by the coordinates of the intersection. The system of equations is //**consistent and independent.**//
 * If the lines overlap (same line), then there are infinitely many solutions, and the system of equations is //**consistent and dependent**//.
 * If the lines do not cross, then there is no solution and the system is **//inconsistent//**.
 * Method 2: Substitution
 * Rearrange one of the equations so a single variable is isolated and expressed as a function of the other variable(s) only. One may then proceed to replace every instance of this variable by the equivalent expression.
 * Ex: {Eqn 1}: x + y = 15 and {Eqn 2}: x + 3y = 35
 * solve {Eqn 1} for y: y = 15 – x
 * replace y in {Eqn 2} with (15 – x): x + 3(15 – x) = 35
 * solve modified {Eqn 2} for x:
 * x + 45 – 3x = 35 leads to x = 5
 * use discovered value to replace x in the other equation and solve for y:
 * 5 + y = 15 leads to y = 10.
 * Check values for both variables in both equations.
 * Method 3: Elimination
 * (more to come -JRH)

// 2-2A Graphing Calculators - Matrices //

 * Guide to basic matrix operations on the TI-83/84 family of calculators:
 * [[file:Basic Matrix Operations.pdf]]

2-2 Introduction to Matrices

 * Here is a list of textbook exercises to try after reading Section 2.2. Students are of course welcome to try others. Answers to all odd-numbered questions may be found in the back of the textbook.
 * 17, 21, 27, 33, 37, 39, 47

2-3 Determinants and Multiplicative Inverses of Matrices
[|Here] is a //geometric// introduction to determinants. Don't be frightened by the vector notation; I'll walk you through it if you come see me.

2-4 Solving Systems of Equations by Using Matrices
[|Here] is a video that demonstrates solving a system of equations using matrices on a TI-83/TI-84 calculator. The lecturer introduces the notion of "reduced row echelon form" (rref) and shows how the rref function is used in solving a system of equations.

Here is a cute problem that illustrates the power of matrices in solving a system of equations.

And here is an interesting application of matrices for coding and decoding messages.

Systems of inequalities can be addressed by graphing.

 * First, treat each inequality as though it were an equation and plot the line (or other function).
 * If the inequality is a strict one (> only or < only), plot the function as a dashed line or curve. Otherwise, use a solid line or curve.
 * For each inequality, select a test point and see if the //(x, y//) pair satisfies the inequality. If so, all the points on that side of the boundary line (or curve) also satisfy the inequality, so shade that whole region lightly. If not, then the other side of the boundary should be shaded.
 * The solution of the system of inequalities is the region that satisfies ALL inequalities, that is, where all the shaded regions overlap.

2-6 Linear Programming

 * Chapter Two Review**
 * A set of review questions: [[file:Chapter Two Test Prep.pdf]]
 * And here are answers: [[file:Chapter Two Test Prep solution.pdf]]