Practice ExercisesPractice Exercises► Perform the indicated operationPerform the indicated operation

1.1. (-2) (-2) ÷÷ (2) (2)2.2. (-6) (-6) ÷÷ (-2) (-2)3.3. (-1) (-1)(-1) (-1)4.4. (-3) ((-3) (++4)4)

5. (0) (-1)5. (0) (-1)6. (-6) – (-6)6. (-6) – (-6)7. (20) – (0)7. (20) – (0)8. (0) (-1)8. (0) (-1)

Problem of the day!!!Problem of the day!!!

► Graph each pair of linear equation on the same set of axes. Graph each pair of linear equation on the same set of axes.

3x – y = - 22x + 2y = -12

y = - ½ x +3y = - ½ x – 1

y + x = 34y + 4x = 12

System of Linear System of Linear EquationEquation

Graphical Graphical SolutionSolution

Types of System of Types of System of Linear Equation in Linear Equation in

Two VariablesTwo Variables

► Independent SystemIndependent System Graph: Intersecting LinesGraph: Intersecting Lines Solution: One SolutionSolution: One Solution

► Inconsistent SystemInconsistent System Graph: Parallel LinesGraph: Parallel Lines Solution: No SolutionSolution: No Solution

►Dependent SystemDependent System Graph: Coincident LinesGraph: Coincident Lines Solution: Infinite number of SolutionSolution: Infinite number of Solution

Activity: Cooperative LearningActivity: Cooperative Learning► Graph the following system, then Graph the following system, then

determine whether the system is determine whether the system is independentindependent, , inconsistentinconsistent, or , or dependentdependent

x + y = 2 x – y = - 1a.

x + 2y = 42x + 4y = 8

b.

y = 3x + 5 y – 3x = - 6 c.

d. 2x + y = 36x + 3y = 9

► Another way to determine Another way to determine whether the system is whether the system is INDEPENDENTINDEPENDENT, , INCONSISTENTINCONSISTENT, or , or DEPENDENTDEPENDENT is by comparing is by comparing the the SLOPESSLOPES and the and the Y–Y–INTERCEPTINTERCEPT of the system of the of the system of the form form y= mx + by= mx + b

I l lustrative Examples:I l lustrative Examples:

DependentDependentthe samethe samethe samethe samey = 3x + 4y = 3x + 4y = 3x + 4y = 3x + 4

InconsistentInconsistentDifferentDifferentthe samethe samey = 2x – 2 y = 2x – 2 y = 2x +3y = 2x +3

IndependentIndependentDifferentDifferentdifferentdifferenty = 3x + 4y = 3x + 4y = 4x – 5 y = 4x – 5

SystemSystemy-intercepty-interceptSlopeSlopeEquationsEquations

Communicating Skil lsCommunicating Skil ls

► When does the system of equation When does the system of equation dependent? Inconsistent? dependent? Inconsistent? Independent?Independent?

► Compare the slopes and the y-Compare the slopes and the y-intercept of each systemintercept of each system

Test ProperTest Proper► Without graphing, determine whether the system is Without graphing, determine whether the system is

dependent, inconsistent, dependent, inconsistent, oror independent. independent.

x = - y + 3x = - y - 5a.

4x + 3y = 23x – 4y = -7

b.

4y + 2x = 5 8y + 4x = 10 c.

d. x = 7 y = 4

INC

IND

DEP

IND

x + 2y = 4 3x + 6y = 12 e. DE

P

AgreementAgreement► Determine whether the system is Determine whether the system is dependent, dependent,

inconsistent, inconsistent, oror independent. independent.

3x + y = 7x = 1a.

8 + 2y = 2x4y + 2x = 8

b.

3x + y = 9 y = 3x + 5 – 6x c.

d. 3x – 6y = 11 2x – 7y = 13

3x – 4y = -5 9x – 12y = 8 e.