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A system of two linear equations may have one solution, an infinite number of solutions, or even no solution at all. Based on the number of solutions, systems of equations can be classified into consistent and inconsistent systems. A consistent system is one that has at least one solution. An independent system is a consistent system that has exactly one solution, whereas a dependent system is a consistent system that has an infinite number of solutions. On the other hand, if a system has no solution, it is referred to as inconsistent. Here, we will be studying how to determine the consistency of a system of linear equations.
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Key Terms: Consistant system, dependent and consistent, system of linear equations, Infinite number, inconsistent systems, linear equations, determinants
Read more: Linear Programming
Consistent System
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A system of linear equations is said to be consistent if there is one solution that satisfies all of the equations.
For example,
If we have the system:
{x + y} = 10
2x + 2y = 20
The solution to both are the line x + y = 10, thus the system is consistent. To make the graph of pair of linear equations in two variables, we draw two lines that represent the equations. The possible cases are:
- There will be a unique solution to the pair of linear equations if both the lines intersect at a point. Here, the pair of linear equations is said to be consistent. In the graph shown above, lines intersect at point P(x,y) which represents the unique solution of the system of linear equations in two variables. Algebraically, if a1/a2 ≠ b1/b2, then the pair of linear equation is consistent.
- Consider two lines with the equations:
a1x + b1y + c1 = 0 and,
a2x + b2y + c2 = 0
Let us assume that these lines coincide/ interacts with each other, then there exist infinitely many solutions as a line consists of infinite points. Here, the pair of linear equations is said to be dependent and consistent. As represented in the graph below, the pair of lines coincides and hence, dependent and consistent.
Discover about the Chapter video:
Determinants Detailed Video Explanation:
Inconsistent System
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An inconsistent system of equations is defined as two or more equations that are impossible to solve on the basis of one set of values for the variables.
For example,
If we have the system,
{x + y} = 10
2x + 2y = 21
On subtracting the second equation from 2 times the first, we get
(2x + 2y = 21) – 2(x + y = 10) →0 = 1.
There are no solutions to this system of equations as the system is false, thus the system is inconsistent.
Read more: Roots of Polynomials
Basic and Free Variables
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The basic variable is defined as any variable that corresponds to a pivot column in the augmented matrix of a system. In other words, a basic variable is bound by an equation.
On the other hand, variables that are not bound by an equation in a matrix, are called free variables.
For example,
3 x 14 x 22 x 41 x 22 x 486 x 1 x 310 →
x1-2x4-31/3x22x48x314x472
In these matrices, the basic variables are
- x1, because it is bound by the equation 2x4 – 31/3.
- x2, because it equals 8 – 2x4.
- x3, because it equals 72 – 14x4.
The free variable here is x4 as it is not bound by any equation.
To determine if a variable is basic or free, check if the variable has a pivot value. In the matrices shown above, the first three columns are pivot columns. It means the three variables x1, x2 and x3 are basic, while the fourth variable x4 is free. If the fifth column or the augmented column, is a pivot column, there is no solution and thus it is inconsistent.
Also check: Pair of Linear Equations in Two Variables Formula
How to Determine if a System of Equations is Consistent
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In order to determine whether a system of linear equations is consistent, the procedure is as follows:
Step 1: Write down the given system of equations in the form of a matrix equation AX = B.
Step 2: Find the rank of A and rank of [A, B] by applying only elementary row operations. (Column operations should not be applied.)
Step 3:
Case 1: If there are n unknowns in the system of equations and
ρ (A) = ρ ([A|B]) = n
Then the system AX = B, is consistent and has a unique solution.
Case 2: If there are n unknowns in the system AX = B
ρ(A) = ρ([A| B]) < n
Then the system is consistent and has infinitely many solutions and these solutions.
Case 3: If ρ(A) ≠ ρ([A| B])
Then the system AX = B is inconsistent and has no solution.
Read more: Calculus Formula
Things to Remember
- When a system of linear equations is inconsistent, it is possible to derive a contradiction from the equations that may always be rewritten as the statement 0 = 1. The graphs of inconsistent equations on the xy – plane 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. The same phenomenon can occur for any number of equations.
- In general, equations are inconstant if the left-hand 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 left-hand sides are linearly independent is always consistent.
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Sample Questions
Ques: What is the consistency theorem in linear algebra? (1 mark)
Ans. According to the consistency theorem, Ax = b is consistent if and only if b is a linear combination of column vectors of A.
Ques: How to find the rank of a matrix? (2 marks)
Ans. The rank of a matrix can be found by counting the number of non-zero rows or non-zero columns. Therefore, if we have to find the rank of a matrix, we will transform the given matrix to its row echelon form and then count the number of non-zero rows.
Ques: How do we know if a system is consistent with a rank? (2 marks)
Ans. We calculate the rank of the coefficient matrix and we calculate the rank of the augmented matrix, if the 2 ranks are the same then it is consistent. If the rank of the coefficient matrix is less than the rank of the augmented matrix then it is inconsistent.
Ques: How do you tell if a system of equations is inconsistent? (2 marks)
Ans. When we graph systems of equations, the intersection of the lines is the solution. If a system has infinitely many solutions, then the lines overlap at every point. In other words, they are the same exact line. This means that any point on the line is a solution to the system.
Ques: Check for the consistency of the following pair of linear equations- (2 marks)
x – 2y = 1
2x – 4y = 2
Ans. To check the condition of consistency we need to find out the ratios of the coefficients of the given equations,
a1a2= 12
b1b2= 12
c1c2= 12
Now, as a1a2= b1b2= c1c2= we can say that the above equations represent lines that are coincident in nature and the pair of equations is dependent and consistent. Therefore, we can say that the lines coincide with each other, having an infinite number of solutions.
Also, when we plot the given equations on a graph, it represents a pair of coincident lines.
Ques: Can graphing be used if the system is inconsistent or independent? (2 marks)
Ans. Yes, in both cases we can still graph the system to determine the type of system and solution. If the two lines are parallel, the system has no solution and is inconsistent. If the two lines are identical, the system has infinite solutions and is a dependent system.
Ques: Can the substitution method be used to solve any linear system in two variables? (1 mark)
Ans. Yes, but the method works best if one of the equations contains a coefficient of 1 or –1 so that we do not have to deal with fractions.
Ques: Solve the following system of equations by graphing. Identify the type of system. (3 marks)
2x + y = -8
x – y = -1
Ans. Solve the first equation for y.
2x+y=−8
y=−2x−8
Solve the second equation for y.
x−y=−1
y=x+1
Graph both equations on the same set of axes:
The lines appear to intersect at the point (−3,−2). We can check to make sure that this is the solution to the system by substituting the ordered pair into both equations.
2(−3) + (−2) = −8
−8=−8 True
(−3) − (−2) = −1
−1=−1 True
The solution to the system is the ordered pair (−3,−2) so the system is independent.
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