Definitions
System of Linear Equations
A set of two or more linear equations
4y-2x=6
8y=4x-12
Inconsistent
4x+2y=9
-4x+3y=16
(-1/4,5)
x+y=7
x+2y=11
(3,4)
x+4y=-1
2x+5y=4
(7,-2)
Equivalent Systems
When you multiply each side of one or both equations in a system by the same nonzero number.
y=2x-3
6x-3y=9
Dependent
-2x+8y=-8
5x-8y=20
(4,0)
x-3y=4
x+2y=1
(11/5, -3/5)
2x+3y-2z=-1
x+5y=9
4z-5x=4
(4,1,6)
Objective Function
The quality you are trying to maximize or minimize
2x+3y=1
4x+y=-3
Independent
2x+7y=4
3x+5y=-5
(-5,2)
x+3y=5
-2x-4y=-5
(-2.5,2.5)
x-6y+4z=-12
x+y-4z=12
2x+2y+5z=-15
(0,0,-3)
Linear Programming
A method for finding a minimum or maximum value of some quantity given a set of constraints
-3x+y=4
x-1/2y=1
Inconsistent
3x+7y=15
5x+2y=-4
(-2,3)
4x-y=20
-2x-2y=10
(3,-8)
2x-y+z=4
x+3y-z=11
4x+y-z=14
(3,3,1)
Types of Systems
Consistent System has at least one solution. Inconsistent system has no solutions. Independent system has one solution. Dependent system has infinitely many solutions.
2x+y=-8
x-y=1
Independent
-4y-11x=36
20=-10x-10y
(-4,2)
3x+4y=12
2x+y=10
(5.6,-1.2)
-x-y-3z=-9
z=-3x-1
x=5y-z+23
(-2,-4,5)