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Chapter 3
Linear Systems
by Troy Cole
# Chapter 3
Linear Systems

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Graphing Systems of Equations

Chapter 3.1

System of equations: a set of two or more equations that use the same variables.

Linear System: a line that contains two identicle variables.

Independent system: has a unique solution.

Dependnt system: does not have a unique solution.

Ex. y= x + 3 y= -2x + 3

Inconsistant system: a system that does not have a solution.

Chapter 3.2

Solving systems Algebraically

Equivalent systems: systems that have the same solution.

Subsitution Ex. 4x + 3y = 4 2x - y = 7

Solve for "y" 2x - y = 7 y= 22 -7

Subsitute the "y" 4x + 3y =4 4x + 3(2x - 7) =4 4x + 6x - 21 =4 4x + 6x =25 x=2.5.

Solve for "y" again y= 2x - 7 y= 2(2.5) - 7 y= -2

Final Answer (2.5, -2)

Chapter 3.3

Systems of Inequalities

Ex. x - 2y < 6 y< -3/2x + 5

Chapter 3.4

Linear Programming

Linear Programming: identifies the minimum and maximum.

Objective Function: is how the minimum is modeled.

Constraints: are the limits on the variables.

Feasible Region: is the area on a graph that satisfies all the constraints.

Chapter 3.5

Graphs in Three Dimension

Coordinate Space: is adding a third axis, "z".

Ordered Triplets: (X,Y,Z)

Trace: when the graph of a pkane intersects one of the coordinate planes in a line.

Chapter 3.6

Systems with Three Variables

Three Systems x - 3y + 3z = -4 2x + 3y - z = 15 4x - 3y - z = 19

Paired for elimination x - 3y + 3z = -4 2x + 3y - z = 15 --------------------- 3x + 2z = 11

and....

Paired for elimination 2x + 3y - z = 15 4x - 3y - z = 19 ---------------------- 6x - 2z = 34

Solve for "x" 3x + 2z = 11 6x - 2z = 34 ----------------- 9x =45 x=5

Solve for "z" 3x + 2z =11 3(5) + 2z = 11 ------------------- 2z = -4 z=-2

Solve by subsitution x - 3y + 3z = -4 5 - 3y + 3(-2) = -4 5 - 3y - 6 = -4 -3y = -3 y=1

Final answer (5,1,-2)