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Inconsistent and Dependent Systems
Two properties of a linear system are consistency (are there solutions? ) and dependency (are the equations linearly independent? ).
Learning Objectives

Recognize when equations in a system are dependent

Recognize when equations in a system are inconsistent
Key Points

The equations of a linear system are independent if none of the equations can be derived algebraically from the others. If a system is not independent, it is dependent.

A linear system is consistent if it has a solution, and inconsistent otherwise.

In general, inconsistencies occur if the lefthand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation.
Term

linear system
A mathematical model of a system based on the use of a linear operator.
Full Text
In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables. For example,
is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of numbers to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by
since it makes all three equations valid.
A linear system may behave in any one of three possible ways:
 The system has infinitely many solutions.
 The system has a single unique solution.
 The system has no solution.
Dependent Systems
The equations of a linear system are independent if none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set. For linear equations, logical independence is the same as linear independence. Systems that are not independent are by definition dependent.
For example, the equations
are dependent — they are the same equation when scaled by a factor of two, and they would produce identical graphs. This is an example of equivalence in a system of linear equations.
For a more complicated example, the equations
are dependent, because the third equation is the sum of the other two. Indeed, any one of these equations can be derived from the other two, and any one of the equations can be removed without affecting the solution set. The graphs of these equations are three lines that intersect at a single point .
Inconsistent Systems
A linear system is consistent if it has a solution, and inconsistent otherwise. When the system is inconsistent, it is possible to derive a contradiction from the equations, that may always be rewritten such as the statement 0 = 1.
For example, the equations
are inconsistent. In fact, by subtracting the first equation from the second one and multiplying both sides of the result by 1/6, we get 0 = 1. The graphs of these equations on the xyplane are a pair of parallel lines .
It is possible for three linear equations to be inconsistent, even though any two of them are consistent together. For example, the equations
are inconsistent. Adding the first two equations together gives 3x + 2y = 2, which can be subtracted from the third equation to yield 0 = 1. Note that any two of these equations have a common solution. The same phenomenon can occur for any number of equations.
In general, inconsistencies occur if the lefthand sides of the equations in a system are linearly dependent, and the constant terms do not satisfy the dependence relation. A system of equations whose lefthand sides are linearly independent is always consistent.
Key Term Reference
 constant
 Appears in this related concepts: Inverse Variation, Combined Variation, and Quadratic Functions of the Form f(x) = a(xh)^2 + k
 equation
 Appears in this related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 factor
 Appears in this related concepts: Factorial Experiments: Two Factors, Finding Factors of Polynomials, and Solving Quadratic Equations by Factoring
 graph
 Appears in this related concepts: Graphing on Computers and Calculators, Reading Points on a Graph, and Graphing Functions
 linear
 Appears in this related concepts: Exponential Growth and Decay, Linear Approximation, and Delivery Tips
 linear equation
 Appears in this related concepts: Linear Equations, Linear Inequalities, and Linear and Quadratic Equations
 parallel lines
 Appears in this related concepts: Parallel and Perpendicular Lines, The Linear Function f(x) = mx + b and Slope, and SlopeIntercept Equations
 point
 Appears in this related concepts: The Intermediate Value Theorem, Graphing Equations, and Introduction: Polynomial and Rational Functions and Models
 relation
 Appears in this related concepts: Symmetry, Visualizing Domain and Range, and Functions and Their Notation
 set
 Appears in this related concepts: Sequences of Statements, Sequences, and Introduction to Sequences
 system of equations
 Appears in this related concepts: The Elimination Method, The Substitution Method, and Models Involving Nonlinear Systems of Equations
 term
 Appears in this related concepts: Basics of Graphing Polynomial Functions, The 22nd Amendment, and Democracy
 variable
 Appears in this related concepts: Calculating the NPV, Controlling for a Variable, and Fundamentals of Statistics
Sources
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Cite This Source
Source: Boundless. “Inconsistent and Dependent Systems.” Boundless Algebra. Boundless, 28 May. 2015. Retrieved 28 May. 2015 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/systemsofequationsandmatrices6/inconsistentanddependentsystems42/inconsistentanddependentsystems1966189/