What is the Condition for a Unique Solution? | Simple Explanation for Class 12

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What is the Condition for a Non-Homogeneous System to Have a Unique Solution?

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

For a non-homogeneous system of linear equations to have a unique solution, the determinant of the coefficient matrix must be non-zero. This means that the lines or planes represented by the equations intersect at exactly one point.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya, buying samosas and chai. Rohan buys 2 samosas and 1 chai for Rs 50. Priya buys 1 samosa and 2 chai for Rs 40. If we want to find the exact price of one samosa and one chai, this system has a unique solution because their purchases are different enough to pinpoint the prices.

Worked Example

Step-by-Step

Let's check if the system x + y = 5 and 2x + 3y = 12 has a unique solution.

Step 1: Write the system in matrix form AX = B.
The coefficient matrix A is [[1, 1], [2, 3]].
The variable matrix X is [[x], [y]].
The constant matrix B is [[5], [12]].

---Step 2: Calculate the determinant of the coefficient matrix A.
det(A) = (1 * 3) - (1 * 2)

---Step 3: Perform the multiplication.
det(A) = 3 - 2

---Step 4: Find the value of the determinant.
det(A) = 1

---Step 5: Check the condition.
Since det(A) = 1, which is not equal to 0, the system has a unique solution.

Answer: The system has a unique solution.

Why It Matters

Understanding unique solutions helps engineers design stable structures and computer scientists create efficient algorithms for AI and machine learning. From predicting stock market trends in FinTech to designing components for EVs, knowing when a system has a single, definite answer is crucial for making accurate decisions and building reliable technology.

Common Mistakes

MISTAKE: Confusing a non-homogeneous system with a homogeneous system. | CORRECTION: Remember, a non-homogeneous system has at least one equation where the constant term (the number on the right side) is not zero. A homogeneous system has all constant terms as zero.

MISTAKE: Forgetting to calculate the determinant correctly, especially for 3x3 matrices. | CORRECTION: Practice calculating determinants carefully. For a 2x2 matrix [[a, b], [c, d]], it's ad - bc. For 3x3, use cofactor expansion along any row or column.

MISTAKE: Assuming a unique solution just because there are enough equations. | CORRECTION: The number of equations and variables is important, but the key condition for a unique solution is that the determinant of the coefficient matrix must be non-zero.

Practice Questions

Try It Yourself

QUESTION: Does the system x + 2y = 7 and 3x + 6y = 21 have a unique solution? | ANSWER: No, because det(A) = 0.

QUESTION: For what value of 'k' will the system 2x + 3y = 5 and 4x + ky = 10 NOT have a unique solution? | ANSWER: k = 6

QUESTION: Consider the system: x - y + z = 4, 2x + y - z = 1, 3x + 2y + 2z = 7. Does this system have a unique solution? (Hint: Calculate the determinant of the 3x3 coefficient matrix). | ANSWER: Yes, det(A) = 15, which is not zero.

MCQ

Quick Quiz

What is the primary condition for a non-homogeneous system of linear equations to have a unique solution?

The determinant of the coefficient matrix is zero.

The determinant of the coefficient matrix is non-zero.

The number of equations is less than the number of variables.

The constant terms are all zero.

The Correct Answer Is:

A unique solution exists when the determinant of the coefficient matrix is non-zero, meaning the equations are independent and intersect at a single point. If the determinant is zero, there are either no solutions or infinitely many solutions.

Real World Connection

In the Real World

When ISRO scientists launch a satellite, they use complex systems of equations to calculate its trajectory. For the satellite to reach its exact orbit, these equations must have a unique solution, ensuring there's only one correct path. Similarly, in FinTech, algorithms predicting stock prices or managing loan approvals rely on systems having unique solutions to give a precise outcome.

Key Vocabulary

Key Terms

Non-homogeneous system: A system of linear equations where at least one constant term is not zero. | Coefficient matrix: A matrix formed by the coefficients of the variables in a system of linear equations. | Determinant: A scalar value calculated from the elements of a square matrix, indicating properties of the linear transformation described by the matrix. | Unique solution: A system of equations having exactly one set of values for the variables that satisfies all equations.

What's Next

What to Learn Next

Now that you understand unique solutions, next explore 'Conditions for No Solution and Infinitely Many Solutions'. This will complete your understanding of all possible outcomes for a system of linear equations and prepare you for more advanced topics!