Inaugurated by IN-SPACe
ISRO Registered Space Tutor
S7-SA2-0142
What is the Consistency of a System of Linear Equations?
Grade Level:
Class 12
AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics
Definition
What is it?
The consistency of a system of linear equations tells us if there are any solutions that satisfy all equations at the same time. If a solution exists, the system is 'consistent'. If no solution exists, the system is 'inconsistent'.
Simple Example
Quick Example
Imagine you go to a chai stall. You buy 2 samosas and 1 chai for Rs 50. Your friend buys 4 samosas and 2 chai for Rs 100. This is a consistent system because there's a price for samosas and chai that makes both statements true (e.g., Samosa Rs 20, Chai Rs 10). If your friend said they bought 4 samosas and 2 chai for Rs 80, this would be inconsistent, as it clashes with your bill.
Worked Example
Step-by-Step
Let's check the consistency of this system:
Equation 1: x + y = 5
Equation 2: 2x + 2y = 10
---Step 1: Write the equations in matrix form AX = B. Here, A = [[1, 1], [2, 2]], X = [[x], [y]], B = [[5], [10]].
---Step 2: Calculate the determinant of matrix A (det(A)). det(A) = (1 * 2) - (1 * 2) = 2 - 2 = 0.
---Step 3: Since det(A) = 0, the system can either have no solution or infinitely many solutions. We need to check the adjoint of A multiplied by B (adj(A)B).
---Step 4: Find adj(A). For a 2x2 matrix [[a, b], [c, d]], adj(A) = [[d, -b], [-c, a]]. So, adj(A) = [[2, -1], [-2, 1]].
---Step 5: Calculate (adj(A))B = [[2, -1], [-2, 1]] * [[5], [10]] = [[(2*5) + (-1*10)], [(-2*5) + (1*10)]] = [[10 - 10], [-10 + 10]] = [[0], [0]].
---Step 6: Since det(A) = 0 AND (adj(A))B = 0, the system has infinitely many solutions.
---Answer: The system is consistent with infinitely many solutions.
Why It Matters
Understanding consistency is crucial for solving real-world problems in many fields. Engineers use it to design stable structures, AI/ML engineers use it to train models, and economists use it to analyze market trends. It helps scientists predict outcomes in physics and climate science.
Common Mistakes
MISTAKE: Assuming det(A) = 0 always means no solution. | CORRECTION: If det(A) = 0, you must check (adj(A))B. If (adj(A))B is not equal to 0, then there is no solution. If (adj(A))B is equal to 0, there are infinitely many solutions.
MISTAKE: Confusing inconsistent with 'unique solution'. | CORRECTION: Inconsistent means NO solution. Consistent means AT LEAST one solution (either unique or infinitely many).
MISTAKE: Incorrectly calculating the determinant or the adjoint matrix. | CORRECTION: Always double-check your calculations for det(A) and adj(A), especially for larger matrices. A small error here will lead to the wrong consistency conclusion.
Practice Questions
Try It Yourself
QUESTION: Is the system x + y = 3 and x - y = 1 consistent? | ANSWER: Yes, it is consistent with a unique solution (x=2, y=1).
QUESTION: Determine the consistency of: x + 2y = 5 and 2x + 4y = 8. | ANSWER: Inconsistent (det(A)=0, but (adj(A))B is not zero).
QUESTION: For what value of 'k' is the system x + ky = 3 and 4x + 8y = 12 consistent with infinitely many solutions? | ANSWER: k = 2.
MCQ
Quick Quiz
If a system of linear equations has det(A) = 0 and (adj(A))B is also equal to 0, what can we say about its consistency?
It is inconsistent.
It is consistent with a unique solution.
It is consistent with infinitely many solutions.
It has no solution.
The Correct Answer Is:
C
When det(A) = 0 and (adj(A))B = 0, the system has infinitely many solutions, meaning it is consistent. Option A and D describe inconsistent systems, and B describes a unique solution (where det(A) is not zero).
Real World Connection
In the Real World
Imagine ISRO scientists tracking multiple satellites. Each satellite's path can be described by linear equations. To make sure their orbits don't collide, they need to check if there's a common 'solution' (a point in space and time) where their paths intersect. If the system is consistent with a unique solution, it means a collision is possible and needs intervention. If inconsistent, their paths won't cross.
Key Vocabulary
Key Terms
Consistent System: A system with at least one solution. | Inconsistent System: A system with no solution. | Unique Solution: Exactly one set of values that satisfies all equations. | Infinitely Many Solutions: An unlimited number of solutions satisfy all equations. | Determinant: A special number calculated from a square matrix.
What's Next
What to Learn Next
Great job understanding consistency! Next, you can explore Cramer's Rule, which is a powerful method to find the unique solution of a consistent system using determinants. This will deepen your understanding of how to actually solve these systems.
