Chapter 1
Section 1.3Consistent Systems of Linear
Equations
{ 3 π₯1β6π₯2+9 π₯3β3 π₯5=18βπ₯1+2π₯2β3π₯3+2π₯4+11π₯5=22π₯1β4 π₯2+6 π₯3β2π₯4β12π₯5=4
ExampleFind the augmented matrix for the system of equations to the right and row reduce it to reduced echelon form.
[ 3β12 β62β4
9β36
02β2
β311β12|
1824 ]
[ 1β12 β22β4
3β36
02β2
β111β12|
624 ]
β R1
[ 100 β200
300
02β2
β110β10|
68β8]
R1+R2
-2R1+R3
Β½R2
[ 100 β200
300
01β2
β15β10|
64β8]
2R2+R3
[ 100 β200
300
010
β150 |640]
Rewriting the equivalent system:
We can rewrite this so that the variables whose coefficient is the leading 1 in row of the matrix in reduced echelon form is on the right side of the equation and all other variables are on the left.
{π₯1=6+2π₯2β3π₯3+π₯5π₯4=4β5 π₯5
Find the solution for:
π₯1=6π₯2=1π₯3=0π₯4=14π₯5=β2
General Solution
Particular Solution
Dependent vs Independent VariablesThe reduced echelon form of an augmented matrix for a consistent system of equations will never have a leading 1 in the last (or augmented) column. Each of the non-augmented columns corresponds to a variable in the system. Each of the columns with a leading 1 the corresponding variable appears on right side when the general solution is expressed.The variables that correspond to the columns with the leading 1βs are the dependent (constrained or determined) variables.The variables that correspond to the columns without the leading 1βs are the independent (unconstrained or free) variables.
[ 100 β200
300
010
β150 |640] {π₯1=6+2π₯2β3π₯3+π₯5π₯4=4β5 π₯5
Dependent Variables:
Independent Variables:
Given values for the independent variables the values for the dependent variables are automatically determined according to the equations in order to form a particular solution. An independent variable can have any value between and . So a system with an independent variable will always have an infinite number of solutions.
ExamplesEach of the matrices below is the augmented matrix for a system of equations. Given the general solution for the system in terms of the dependent variables and tell which variables are dependent and which are independent.
[1000
0100
β5300
0β100
0010
3720]
Matrix VariablesGeneral Solution
{ ππ=π+π ππ
ππ=πβπ ππ+ππ
ππ=π
Dependent:
Independent:
[0000
1000
2000
0100
3β200
0000
2β500
]{π₯2=2β2π₯3β3 π₯5π₯4=β5+2π₯5
Dependent:
Independent:
[1000
3000
0100] No Solution the system
is not consistent
[1 β4 20 0 00 0 0
000 ] {π₯1=4 π₯2β2 π₯3
Dependent:
Independent:
Solutions to linear systemsThe number of solutions to a linear system only has three possible numbers; None, a unique (only one) solution, or an infinite number of solutions.
1. The system has none when it is inconsistent.2. The system has a unique solution when it is
consistent but has no independent variables (i.e. all variables are dependent).
3. The system has infinitely many solutions when it consistent but has at least one (maybe more) independent variables.
[1 0 β20 1 00 0 0
501] Inconsistent
No Solutions
[1 0 00 1 00 0 1
50β2] No
Independent Variable
One Solution
[1 0 β20 1 10 0 0
500]
ConsistentInfinite Solutions
Homogeneous Systems of EquationsA homogeneous system of linear equations is a system where all the constants are zero. In matrix form the augmented (or last) column is all zeros. Because of this a homogeneous system is always consistent. It always has the solution of having all variables being zero. If there are no independent variable this solution is unique (only one) if there is an independent variable there is an infinite number of solutions.
Always has solution: