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(6.4) Non-UniqueSolutions

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Defining a System of Equations

A grouping of 2 or more equations,containing one or more variables.

x + y = 2x + y = 2

2x + y = 52x + y = 5

2y = x + 22y = x + 2

y = 5xy = 5x -- 77

6x6x -- y = 5y = 5

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How do we ´solveµ asystem of equations???

By finding the point where two or more

equations, intersect.

x + y = 6x + y = 6

y = 2xy = 2xPoint of intersectionPoint of intersection

66

44

22

11 22

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How do we ´solveµ asystem of equations???

By finding the point where two or more

equations, intersect.

x + y = 6x + y = 6

y = 2xy = 2x(2,4)(2,4)

66

44

22

22

11

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ax + by = c

2x + 3y = 62x + 3y = 6 ax + by = cax + by = c

--2x2x --2x2x3y = 63y = 6 -- 2x2x33 33 33

y = 2y = 2 -- 2233 xx

y =y = -- + 2+ 222

33xx y = mx + by = mx + b

WE WANT THIS FORM!!!WE WANT THIS FORM!!!

(Standard Form)(Standard Form)

(Slope(Slope-- Intercept)Intercept)

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ax + by = c

NOTE: The equationNOTE: The equation ax + by = cax + by = c is just another is just another 

form of a linear equation.form of a linear equation.

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Non-Unique Solutions

No SolutionNo Solution:: when lines of a graph are parallelwhen lines of a graph are parallel

have same slope and different yhave same slope and different y--interceptsintercepts

since they do not intersect, there is no solutionsince they do not intersect, there is no solution

also called analso called an Inconsistent SystemInconsistent System

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Infinite SolutionsInfinite Solutions::

Non-Unique Solutions

a pair of equations that have the samea pair of equations that have the same

slope and yslope and y--intercept.intercept.

also call aalso call a Dependent SystemDependent System

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Non-Unique Solutions

One SolutionOne Solution:: the lines of two equations intersectthe lines of two equations intersect

also called analso called an Independent SystemIndependent System

have different slope and same or different yhave different slope and same or different y--

interceptsintercepts

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Examples«

1)1)

Determine whether the following dependent, independent,Determine whether the following dependent, independent,

or inconsistent by looking at the slope and yor inconsistent by looking at the slope and y--interceptsintercepts

2y + x = 82y + x = 8

y = 2x + 4y = 2x + 4

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ax + by = c

2y + x = 82y + x = 8 ax + by = cax + by = c

--xx --xx2y = 82y = 8 -- xx22 22 22

y = 4y = 4 -- 1122 xx

y =y = -- + 4+ 411

22xx y = mx + by = mx + b

WE WANT THIS FORM!!!WE WANT THIS FORM!!!

(Standard Form)(Standard Form)

(Slope(Slope-- Intercept)Intercept)

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y =y = -- + 4+ 411

22xx slope =slope = --

yy--intercept = 4intercept = 4

11

22

y = 2x + 4y = 2x + 4 slope = 2slope = 2

yy--intercept = 4intercept = 4

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Examples«

1)1)

Determine whether the following dependent, independent,Determine whether the following dependent, independent,

or inconsistent by looking at the slope and yor inconsistent by looking at the slope and y--interceptsintercepts

2y + x = 82y + x = 8

y = 2x + 4y = 2x + 4

 ANS: ANS:IndependentIndependent

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Examples«Determine whether the following dependent, independent,Determine whether the following dependent, independent,

or inconsistent by looking at the slope and yor inconsistent by looking at the slope and y--interceptsintercepts

2)2) y =y = --6x + 86x + 8

y + 6x = 8y + 6x = 8

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ax + by = c

y + 6x = 8y + 6x = 8 ax + by = cax + by = c

--6x6x --6x6xy = 8y = 8 -- 6x6x

y =y = -- 6x + 86x + 8

y = mx + by = mx + b

WE WANT THIS FORM!!!WE WANT THIS FORM!!!

(Standard Form)(Standard Form)

(Slope(Slope-- Intercept)Intercept)

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slope =slope = --66

yy--intercept = 8intercept = 8

y =y = --6x + 86x + 8

slope =slope = --66

yy--intercept = 8intercept = 8y =y = -- 6x + 86x + 8

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Examples«Determine whether the following dependent, independent,Determine whether the following dependent, independent,

or inconsistent by looking at the slope and yor inconsistent by looking at the slope and y--interceptsintercepts

2)2) y =y = --6x + 86x + 8

y + 6x = 8y + 6x = 8

 ANS: ANS:DependentDependent

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Examples«Determine whether the following dependent, independent,Determine whether the following dependent, independent,

or inconsistent by looking at the slope and yor inconsistent by looking at the slope and y--interceptsintercepts

3)3) xx -- 5y = 105y = 10

--5y =5y = --x +6x +6

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ax + by = c

xx ± ± 5y = 105y = 10 ax + by = cax + by = c

--xx --xx--5y = 105y = 10 -- xx--55 --55 --55

y =y = --2 +2 + 1155 xx

y =y = -- 2211

55xx y = mx + by = mx + b

WE WANT THIS FORM!!!WE WANT THIS FORM!!!

(Standard Form)(Standard Form)

(Slope(Slope-- Intercept)Intercept)

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ax + by = c

--5y =5y = --x + 6x + 6 ax + by = cax + by = c

--55 --55 --55

66

55y =y = --

11

55xx

y = mx + by = mx + b

WE WANT THIS FORM!!!WE WANT THIS FORM!!!

(Standard Form)(Standard Form)

(Slope(Slope-- Intercept)Intercept)

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slope =slope =

yy--intercept =intercept = --22

slope =slope =

yy--intercept =intercept = --

y =y = -- 2211

55xx

11

55

66

55y =y = --11

55 xx

11

55 66

55

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Examples«Determine whether the following dependent, independent,Determine whether the following dependent, independent,

or inconsistent by looking at the slope and yor inconsistent by looking at the slope and y--interceptsintercepts

3)3) xx -- 5y = 105y = 10

--5y =5y = --x +6x +6

 ANS: ANS:InconsistentInconsistent