Chapter 1

Section 1.3Consistent Systems of Linear

Equations

{ 3 𝑥1−6𝑥2+9 𝑥3−3 𝑥5=18−𝑥1+2𝑥2−3𝑥3+2𝑥4+11𝑥5=22𝑥1−4 𝑥2+6 𝑥3−2𝑥4−12𝑥5=4

ExampleFind the augmented matrix for the system of equations to the right and row reduce it to reduced echelon form.

[ 3−12 −62−4

9−36

02−2

−311−12|

1824 ]

[ 1−12 −22−4

3−36

02−2

−111−12|

624 ]

⅓R1

[ 100 −200

300

02−2

−110−10|

68−8]

R1+R2

-2R1+R3

½R2

[ 100 −200

300

01−2

−15−10|

64−8]

2R2+R3

[ 100 −200

300

010

−150 |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.

{𝑥1=6+2𝑥2−3𝑥3+𝑥5𝑥4=4−5 𝑥5

Find the solution for:

𝑥1=6𝑥2=1𝑥3=0𝑥4=14𝑥5=−2

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 1’s are the dependent (constrained or determined) variables.The variables that correspond to the columns without the leading 1’s are the independent (unconstrained or free) variables.

[ 100 −200

300

010

−150 |640] {𝑥1=6+2𝑥2−3𝑥3+𝑥5𝑥4=4−5 𝑥5

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

−5300

0−100

0010

3720]

Matrix VariablesGeneral Solution

{ 𝒙𝟏=𝟑+𝟓 𝒙𝟑

𝒙𝟐=𝟕−𝟑 𝒙𝟑+𝒙𝟒

𝒙𝟓=𝟐

Dependent:

Independent:

[0000

1000

2000

0100

3−200

0000

2−500

]{𝑥2=2−2𝑥3−3 𝑥5𝑥4=−5+2𝑥5

Dependent:

Independent:

[1000

3000

0100] No Solution the system

is not consistent

[1 −4 20 0 00 0 0

000 ] {𝑥1=4 𝑥2−2 𝑥3

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 −20 1 00 0 0

501] Inconsistent

No Solutions

[1 0 00 1 00 0 1

50−2] No

Independent Variable

One Solution

[1 0 −20 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: