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Mathematics

MATH 1107

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05/09/2013
Different Solutions?
Connect: Linear Systems in High School:
One Unique Solution: No solution: Infinitely many solutions:
Different Solutions With Higher Orders
Solution to a System:
A solution to a linear system is an ordered(x1,x 2x 3...,xn) such that
when you substitute the values into every equation, equality will hold.
Example: (3,2,1) is a solution to the following:
3x −2x + x = 6
1 2 3 Sub x1= 3, x2= 2, 6 = 6
− x1+ x 2 x =30 and x = 1 0 = 0
2 x + x + x =2 12 3
1 2 3 12 =12
Solution Set:
A set of all possible solutions that will solve the system. We use the notation
as follows:S ={list of solutions}
1 05/09/2013
Finding Solutions
To find a solution to a linear system, we use elimination to continuously
reduce the number of variables:
x −2x +3x = 6 x −2x +3x = 6
1 2 3 E2▯(E2)-2(E1) 1 2 3 E2÷5
2x1+ x 2 x =37 5x 25x = −3
E3▯(E1)+(E3)
− x1+ x 2 x =3−2 − x2+2x = 3
x −2x +3x = 6 x −2x +3x = 6
1 2 3 E3▯(E3)+(E2) 1 2 3
x2− x 3 −1 x2− x 3 −1
− x 22x = 3 x3= 3
Sub into E2
x3= 3 x 23= −1 Sub into E1 x1−2(2)+3(3) = 6
x = 2 x +5 = 6
2 1
x1=1
The solution isS ={(1,2,3)}
No Solutions
Consider the following system:
x1−2x +2x = 6 3 E2▯(E2)-2(E1) x1−2x +32 = 6 3
2x + x + x = 7 5x −5x = −5 E2÷5
1 2 3 2 3 E3÷5
3x − x +4x = −2 E3▯-3(E1)+(E3) 5x −5x = −20
1 2 3 2 3
x −2x +3x = 6
1 2 3 E3▯(E3)-(E2) x1−2x +32 = 6 3
x − x = −1
2 3 x2− x 3 −1
x2− x 3 −4 0 = −3
There is no Solution!
2 05/09/2013
Infinite Solutions
Consider the following system:
x −2x +3x = 6 x −2x +3x = 6
1 2 3 E2▯(E2)-2(E1) 1 2 3 E2÷5
2x 1 x +2x = 3 5x 25x = −3
E3▯-3(E1)+(E3) E3÷5
3x 1 x +2x =133 5x 25x = 35
x1−2x +2x = 6 3 x −2x +3x = 6
E3▯(E3)-(E2) 1 2 3
x 2 x =3−1 x2− x 3 −1
x − x = −1
2 3 0 = 0
x3can be what ever we want! This means x =3t, where t can be any number
you want it to be. Substituting this back into the equations we get:
Sub into E2
x 3 t x2−t = −1 Sub into E1 x1−2(−1+t)+3(t) = 6
x +t +2 = 6
x2= −1+t 1
x = 4−t
1
The solution is S ={(4−t,−1+t,t):t∈R}
Different Types of Solutions
Consistent System:
A system that has a solution (either one or many).
Inconsistent System:
A system that has no sol

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