How to Check if a System of Equations is Consistent or Inconsistent?
The concept of consistent and inconsistent systems plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. It is especially important in algebra, where students learn to solve systems of equations and must decide if those systems have solutions or not.
What Is Consistent and Inconsistent Systems?
A consistent system is a system of equations that has at least one solution. In contrast, an inconsistent system is one where no set of variable values will satisfy all the equations simultaneously. You’ll find these concepts applied in algebra (solving simultaneous equations), geometry (analyzing lines on a graph), and matrices (checking systems in higher dimensions).
| Type of System | Solution | Example |
|---|---|---|
| Consistent | At least one solution (unique or infinite) |
x + y = 6, x – y = 2 |
| Inconsistent | No solution | x – y = 8, 5x – 5y = 25 |
Consistent and Inconsistent Systems in Class 10 Maths
In Class 10, consistent and inconsistent systems usually appear as pairs of linear equations in two variables. These can be graphed as straight lines. Here’s a quick summary of meanings:
- If the lines intersect at a single point → Consistent (unique solution)
- If the lines are parallel → Inconsistent (no solution)
- If the lines coincide (overlap exactly) → Consistent and Dependent (infinite solutions)
Key Formula for Consistency of Linear Equations
For two equations:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
Here’s the test:
-
Consistent System:
If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \): unique solution (lines intersect)If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \): infinite solutions (lines coincide)
-
Inconsistent System:
If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \): no solution (lines are parallel)
How to Check Consistency: Step-by-Step Example
Let’s test these two equations:
2x + 3y = 7
4x + 6y = 15
2x + 3y – 7 = 0
4x + 6y – 15 = 0
2. Find ratios:
\( \frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2} \)
\( \frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2} \)
\( \frac{c_1}{c_2} = \frac{-7}{-15} = \frac{7}{15} \)
3. Check the condition:
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \) but \( \neq \frac{c_1}{c_2} \) → the system is inconsistent
4. Final Answer: No solution; the lines are parallel.
Consistent, Inconsistent, and Dependent Systems: Quick Table
| Type | Number of Solutions | Graph Form | Example |
|---|---|---|---|
| Consistent & Independent | One | Intersecting Lines | x + y = 6 x – y = 2 |
| Consistent & Dependent | Infinite | Coincident Lines | x + y = 6 2x + 2y = 12 |
| Inconsistent | None | Parallel Lines | 2x + y = 3 4x + 2y = 8 |
Visualizing Consistent and Inconsistent Systems
- Intersecting lines: Consistent (unique solution)
- Parallel lines: Inconsistent (no solution)
- Coinciding lines: Consistent and dependent (infinite solutions)
Use graph paper or Vedantu’s Graphical Representation of Data resource to see these lines plotted.
Common Student Mistakes
- Mixing up the ratio tests (using c1 and c2 incorrectly)
- Thinking parallel lines are consistent
- Not converting equations to the correct form before comparing coefficients
Real-Life and Cross-Disciplinary Usage
Consistent and inconsistent systems are useful in Maths, Physics (solving forces, circuits), and Computer Science (solving systems in algorithms). Consistent systems ensure unique answers; inconsistent systems warn us of contradictions in real-world conditions.
Short Trick to Check Consistency
A quick classroom method: Check the ratios of x and y, then c.
If the ratio of x and y is different, the system is consistent with one solution. If all ratios match, infinite solutions (dependent). If x and y ratios match but c is different, inconsistent—no solution. Vedantu teachers use this rapid trick in live revision sessions.
Try These Yourself
- Test if the system: 3x – 2y = 7 and 6x – 4y = 14 is consistent or inconsistent.
- Give an example of an inconsistent system of two equations.
- Graph y = 2x + 1 and y = 2x – 3. Are they consistent?
- Make up your own pair of consistent, dependent equations.
Relation to Other Concepts
The concept of consistency connects closely to simultaneous equations, matrix solutions, and linear equations. Understanding it helps you in chapters on inequalities, determinants, and application word problems.
Classroom Tip
A simple way to remember: Look at the graph! If lines cross—consistent. If lines overlap—consistent and dependent. If lines never meet—parallel—which means inconsistent. Try practicing with graph sheets or with Vedantu’s graph resources.
We explored consistent and inconsistent systems—from definitions, formula, exam tips, mistakes, and connections to other subjects. Keep practicing consistency checks with sample questions and worksheets from Vedantu to master this concept!
Explore these for deepening your understanding:
FAQs on Consistent and Inconsistent Systems Explained for Class 10 Maths
1. What is the difference between a consistent and an inconsistent system of equations?
A consistent system of equations has at least one solution, meaning there's at least one set of values for the variables that satisfies all equations simultaneously. An inconsistent system has no solution; there are no values that satisfy all equations.
2. How can you check if a system is consistent or inconsistent?
Several methods exist to determine consistency. One common approach involves comparing the ratio of coefficients and constants. Another method utilizes matrices and their rank. If the rank of the coefficient matrix equals the rank of the augmented matrix, the system is consistent. Otherwise, it's inconsistent. Graphically, intersecting or coincident lines represent consistent systems, while parallel lines indicate an inconsistent system.
3. Are parallel lines considered consistent or inconsistent systems?
Parallel lines represent an inconsistent system. Because parallel lines never intersect, there is no point (x, y) that satisfies both equations simultaneously; hence, no solution exists.
4. What is a dependent system in linear equations?
A dependent system of linear equations is a consistent system with infinitely many solutions. Graphically, this is represented by coincident lines (lines that overlap completely). Algebraically, one equation can be obtained by multiplying the other by a constant.
5. Does every consistent system have a unique solution?
No. A consistent system can have either a unique solution (one solution) or infinitely many solutions (a dependent system). A unique solution occurs when the lines representing the equations intersect at a single point.
6. Can a system be consistent and have more than one solution?
Yes. If the equations are dependent (representing coincident lines), the system is consistent with infinitely many solutions. This means there are many sets of values satisfying all equations.
7. How does the rank of a matrix determine consistency?
The rank of a matrix indicates the number of linearly independent rows or columns. A system of linear equations is consistent if the rank of the coefficient matrix is equal to the rank of the augmented matrix. If the ranks are different, the system is inconsistent.
8. What is the real-life application of consistent & inconsistent systems?
Consistent and inconsistent systems find application in various fields, including engineering, network analysis, and scientific modeling. They help determine if multiple conditions or constraints can be simultaneously satisfied (consistent) or if there's a conflict (inconsistent).
9. What happens if all coefficients are zero?
If all coefficients of the variables in a system of equations are zero, the equations reduce to the form 0 = c, where 'c' is a constant. If c is also zero (0 = 0), the system is dependent and consistent, with infinitely many solutions. If c is non-zero (e.g., 0 = 5), the system is inconsistent (no solution).
10. Are coincident lines always consistent?
Yes, coincident lines always represent a consistent and dependent system. Since the lines overlap entirely, there are infinitely many common points (solutions) that satisfy both equations.
11. Can inconsistent systems exist in non-linear equations?
Yes, inconsistency can occur in both linear and non-linear systems. It arises when there are contradictory conditions or requirements that cannot be simultaneously satisfied by any values of the variables.
12. How do I prove consistent linear equations are true?
To prove a system of linear equations is consistent, demonstrate that the ranks of both the coefficient matrix and the augmented matrix are equal. This can be done using elementary row operations to reduce the augmented matrix to its row-echelon form. A homogeneous system (where all constants are zero) is always consistent.
