Unit+3

=Unit 3 Linear Systems=

Vocabulary

 * constraints (p. 135) || inconsistent system (p. 118) || ordered triples (p. 142) ||
 * coordinate space (p. 142) || independent system (p. 118) || system of equations (p. 116) ||
 * dependent system (p. 118) || linear programming (p. 135) || trace (p. 144) ||
 * equivalent systems (p. 125) || linear system (p. 116) ||  ||
 * feasible region (p. 136) || objective function (p. 135) ||

3-1 Objectives
A **system of equations** is a set of two or more equations that use the same variables. The points where all the graphs intersect represent solutions. You must check the coordinates of the points of intersection in the original equations to be sure you have a solution. A **linear system** consists of linear equations. An **independent system** has a unique solution while a **dependent system** does not have a unique solution. An **inconsistent system** has no solutions.
 * 1) To solve a system by graphing (p. 116)

3-2 Objectives
If you can easily solve one equation in a system of two equations for one of the variables, you can substitute that expression in the other equation. Then you can find the value of the other variable. Otherwise, you can multiply one or both equations by a nonzero quantity to create two terms that are additive inverses. This creates an **equivalent system** of equations. Adding the two equations then eliminates one variable. Again, you can solve for the other variable. In either case, you substitute the value of this second variable into either of the original equations to find the value of the first variable. Recall that some systems have an infinite number of solutions and some have no solutions.
 * 1) To solve a system by substitution (p. 123)
 * 2) To solve a system by elimination (p. 124)

3-3 Objectives
The solution of a system of inequalities is represented on a graph by the region of overlap of the inequalities. To solve a system by graphing, first graph the boundaries for each inequality. Then shade the regions of the plane containing the solutions for both inequalities.
 * 1) To solve systems of linear inequalities (p. 130)

3-4 Objectives

 * 1) To find maximum and minimum values (p. 135)
 * 2) To solve problems with linear programming (p. 137)
 * Linear programming** is a technique used to find the maximum or minimum value of an **objective function**. Linear inequalities are **constraints** on the variables of the objective function. The solutions to the system of constraints are contained in the **feasible region**. The maximum or minimum value of the objective function occurs at a vertex of the feasible region.

3-5 Objectives
You can plot **ordered triples** in **coordinate space**. To sketch a plane that is the graph of an equation in three variables, find the intercepts. To find the //x//-intercept, substitute 0 for //y// and //z//. Then find the other two intercepts. If the plane does not pass through the origin, connect the resulting intercepts on the three axes. These lines are called the **traces** of the plane.
 * 1) To graph points in three dimensions (p. 142)
 * 2) To graph equations in three dimensions (p. 144)

3-6 Objectives
You can solve systems of three equations in three variables using the technique of substitution you learned in [|Lesson 3-2.] Elimination with three equations in three variables involves pairing the equations. Use one equation twice. Then eliminate the same variable in both pairs. The result is a system of two equations in two variables. Proceed using the methods you learned in [|Lesson 3-2.]
 * 1) To solve systems in three variables by elimination (p. 148)
 * 2) To solve systems in three variables by substitution (p. 151)