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Inconsistent System occurs when the lines or planes created by the systems of equations do not intersect at any point or are not parallel. A linear or nonlinear system of equations is said to be constant if at least one set of unknown values satisfies each equation in the system. This happens when they are substituted into each equation, they make each equation remain true as an identity. In contrast, a linear or nonlinear equation system is said to be inconsistent if no collection of unknown values fulfills all of the equations.
Also Read: Consistent Systems of Linear Equations
| Table of Contents |
Key Terms: Consistent Equations, Inconsistent Equations, Dependent System, Free Variables, Elimination Technique, Planes, Lines, Equation system
Consistent and Inconsistent Equations
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A system of equations that is consistent has at least one solution. If you have the system, you can do the following:
The Above Equation is consistent because the solution is in line x+y=10
If an equation system has no solutions, it is inconsistent. It is inconsistent if the final column (in an augmented matrix) is a pivot column, that is, if it contains a pivot.
If you remove the second equation from 2 times the first, you get inconsistency. As a result, the system is untrue, and there are no solutions to the system of equations.
Consistent and Inconsistent Systems
Discover about the Chapter video:
Determinants Detailed Video Explanation:
Check Important Notes for Determinant of a Matrix
Derivating Inconsistency of Equation
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The steps to derive the Inconsistent Equation is as follows:
Create a matrix equation AX = B from the following system of equations.
- Step 1: Determine the system of equations' augmented matrix [A, B].
- Step 2: Using just basic row operations, determine the rank of A and the rank of [A, B]. Column operations should not be used.
- Step 3 is as follows:
- Case 1: If the system of equations has n unknowns and
ρ(A) = ρ([A|B]) = n
The system AX = B, on the other hand, is consistent and has a unique solution.
- Case 2: If the system has n unknowns, AX = B.
ρ(A) = ρ([A| B]) < n
The system is therefore consistent and has an endless number of solutions, and these solutions.
- Case 3:
ρ(A) ≠ ρ([A| B])
The system AX = B is therefore inconsistent and has no solution.
Also Read:
| Related Articles | ||
|---|---|---|
| Methods of Integration | Integration by Partial Fractions | Integration by Parts |
| Fraction | Consistent System | Applications of Determinants and Matrices |
Dependent System
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We know from dealing with two-variable systems of equations that a dependent system of equations has an unlimited number of solutions. The same holds true for three-variable dependent systems of equations. Several scenarios can lead to an unlimited number of solutions.
Three planes might be the same, therefore the answer to one equation will also be the solution to the other two. All three equations might be different, but they all meet on a line that contains an unlimited number of solutions. Alternatively, two of the equations might be the same and cross the third on a straight line.
Dependent System
Read Further: Determinant Formula
Examples of Inconsistent System
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Here are two examples of Inconsistent System:
Undetermined and Inconsistent
The procedure
x + y + z = 3,
x + y + z =4
has no solutions, as demonstrated by subtracting the first equation from the second to yield the illogical 0 = 1.
The system is non-linear.
x2 + y2 + z2 = 17,
x2 + y2 + z2 = 14
has no answers because subtracting one equation from the other yields the impossible 0=3
Check More: Concept of Elementary Row and Column Operators
Determined & Inconsistent System
The procedure
x + y = 3,
4x + 4y = 10
There are no solutions; the contradiction may be demonstrated by multiplying the first equation by 4 and subtracting the second equation to get the impossible 0 = 2.
Likewise,
x3 + y3 + z3 = 10,
x3 + 2y3 + z3 = 12,
3x3 + 5y3 + 3z3 = 32
Because the first equation plus twice the second minus the third contains the contradiction 0 = 2, the system is inconsistent.
Also Read: Determinants
Basic and Free Variables
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A fundamental variable is one that is restricted by an equation. A free variable is one that is not constrained by any equation. Here's an illustration:
Basic and Free Variables
Check whether a variable has a pivot value to see if it is basic or free. The first, second, and third columns in the matrix above were pivot columns, indicating that those three variables were fundamental, while the fourth was free. If the fifth column, or augmented column, is a pivot column, the solution is incoherent, and there is no solution at all.
Also Read: Area of a Triangle
Elimination Technique
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Elimination technique is used to remove the remaining variables in order to solve the variable in a system of equations. This method of elimination is also known as elimination by addition. So, after finding the values for the other variables, the right value for the other variable is swapped into the original equation.
The steps of the Elimination Method are as follows:
- Rewrite the equations to align the variables.
- Change one of the equations such that when the equations are combined together.
- In both equations, a variable will cancel itself out.
- Equations must be inserted in order to remove the variable.
- Solve the last variable.
- Back-substitute the previous variable to solve for the other variable.
Also Read: Substitution Method of Solving a Pair of Linear Equations
Things to Remember
- Inconsistency in a three-variable system of equations, which indicates it lacks a solution that satisfies all three equations.
- Equations might represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect but not at the same point as the other two.
- A linear system is consistent if and only if its coefficient matrix and augmented matrix have the same rank (the coefficient matrix with an extra column added, that column being the column vector of constants).
- A two-variable system of equations is defined as equations of two lines, and they can have an endless number of solutions if the two lines are parallel and can be written as multiples of each other. This is a simple approach to identify systems with an unlimited number of solutions.
- The elimination technique is used to remove the remaining variables in order to solve the variable in a system of equations. This method of elimination is also known as elimination by addition. So, after finding the values for the other variables, the right value for the other variable is swapped into the original equation.
- If an equation system is inconsistent, it is possible to manipulate and mix the equations in such a manner that conflicting information is obtained, such as 2 = 1.
Also Read:
Sample Questions
Ques. What is the solution set to an inconsistency in an equation system? (2 Marks)
Ans. A solution set is the collection of all the intersection points of the system's equations. A solution set may include a finite number of solutions, an infinite number of solutions, or no solutions at all. When there is no solution, the system is said to be inconsistent.
Ques. Is it possible for an overdetermined system to remain consistent? (2 Marks)
Ans. When overdetermined systems are built with irregular coefficients, they are inherently inconsistent. However, they are not necessarily consistent; for example, when certain equations are linear combinations of the other equations in the system. As a result, a predetermined system can be consistent.
Ques. What is the inconsistency formula? (3 Marks)
Ans. Inconsistent equations are defined as two or more equations that are difficult to solve when just one set of variables is used. x+2=4 and x+2=6 are two examples of incorrect equations.
If the system is inconsistent, you will eventually come upon a statement that makes no sense, such as 0 = 3. If this happens, you'll end up with inconsistency in your equations.
Ques. How can you tell whether an equation system is inconsistent? (3 Marks)
Ans. It is dependent on whether a consistent system has an endless number of solutions. When the equations are graphically represented, they depict a similar line. When a system does not have a solution, it is said to be inconsistent:
- It is dependent if a consistent system has an endless number of solutions.
- Both equations reflect the same line when plotted on a graph.
Ques. What exactly is the substitution method? (3 Marks)
Ans. The algebraic method for solving simultaneous linear equations is the substitution method. The meaning of one variable from one equation is substituted in the second equation, as the name implies. A pair of linear equations is therefore reduced into a single linear equation with only one variable, which may subsequently be solved quickly. The steps are as follows:
- Expand the parenthesis to simplify the given equation.
- For either x or y, solve one of the equations.
- In the other equation, substitute the step 2 solution.
- Now, using simple arithmetic processes, answer the new equation.
- Lastly, solve the equation to determine the second variable's value.
Ques. How do you determine an equation's consistency? (3 Marks)
Ans. If both lines cross at the same place, then the pair of linear equations has a unique solution. The pair of linear equations is seen to be consistent in such situations. Keep the following in mind:
- A system is said to be constant if it has at minimum one solution.
- It is autonomous if a consistent system has precisely one solution.
- It is dependent if a consistent system has an endless number of solutions.
- Both equations reflect the same line when plotted on a graph.
Ques. What is the difference between the Substitution method & the Elimination method? (3 Marks)
Ans. The substitute method entails solving an equation to determine the variable value, which is then substituted in another equation. The elimination method, on the other hand, is the act of removing variables from an equation such that the system of equations can be reduced to a single variable.
The main difference between the replacement and elimination methods is that the substitution approach involves substituting a value for the variable, whilst the elimination method involves deleting the variable from the set of linear equations.
Ques. What is the system of equations? (4 Marks)
Ans. A system of equations is a grouping of two or more equations that all have the same variables. When you're dealing with a challenge involving multiple unknown quantities, these systems come in helpful.
Let's say we're seeking two numbers, 5 times the first number multiplied by 2 equals the second number, and 2 times the second number is subtracted from 10 times the first number equals 12.
We have two unknowns in this scenario. Let's say x is the first number and y is the second. We know that adding 5 times the first number to 2 equals the second number;
hence, 5x + 2 = y.
We're also told that subtracting 2 times the second number from ten times the first gives us 12; hence, 10x - 2y = 12.
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