8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 1/22
(6.4) Non-UniqueSolutions
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 2/22
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
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 3/22
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
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 4/22
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
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 5/22
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)
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 6/22
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.
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 7/22
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
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 8/22
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
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 9/22
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
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 10/22
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
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 11/22
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)
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 12/22
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
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 13/22
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
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 14/22
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
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 15/22
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)
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 16/22
slope =slope = --66
yy--intercept = 8intercept = 8
y =y = --6x + 86x + 8
slope =slope = --66
yy--intercept = 8intercept = 8y =y = -- 6x + 86x + 8
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 17/22
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
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 18/22
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
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 19/22
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)
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 20/22
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)
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 21/22
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
8/8/2019 (6.4) Non-Unique Solutions
http://slidepdf.com/reader/full/64-non-unique-solutions 22/22
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
Top Related