What is Cramer's Rule for Unique Solution? | Simple Explanation for Class 12

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What is the Cramer's Rule for 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?

Cramer's Rule is a clever method that uses determinants to solve a system of linear equations. For a unique solution, it helps us find the exact values of variables like 'x', 'y', and 'z' when we have the same number of equations as variables, and the main determinant is not zero.

Simple Example

Quick Example

Imagine you go to a chai stall. One day, 2 cups of chai and 1 samosa cost Rs. 40. Another day, 1 cup of chai and 2 samosas cost Rs. 35. Cramer's Rule can help you quickly find the exact price of one chai and one samosa separately, without guessing!

Worked Example

Step-by-Step

Let's solve this system of equations using Cramer's Rule:
Equation 1: x + y = 5
Equation 2: 2x - y = 1

Step 1: Write the equations in matrix form and find the determinant of the coefficient matrix (D).
D = | 1 1 |
| 2 -1 | = (1 * -1) - (1 * 2) = -1 - 2 = -3

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Step 2: Find Dx by replacing the x-column with the constant terms.
Dx = | 5 1 |
| 1 -1 | = (5 * -1) - (1 * 1) = -5 - 1 = -6

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Step 3: Find Dy by replacing the y-column with the constant terms.
Dy = | 1 5 |
| 2 1 | = (1 * 1) - (5 * 2) = 1 - 10 = -9

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Step 4: Calculate x and y using the formulas: x = Dx / D and y = Dy / D.
x = -6 / -3 = 2
y = -9 / -3 = 3

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Answer: The unique solution is x = 2 and y = 3.

Why It Matters

Understanding Cramer's Rule is like learning a shortcut for solving puzzles with multiple unknowns. Engineers use it to design circuits, economists use it to model market prices, and even in AI/ML, similar math helps computers learn patterns. It's a foundational skill for many exciting careers!

Common Mistakes

MISTAKE: Students forget to check if the main determinant (D) is zero. If D = 0, Cramer's Rule cannot be used for a unique solution. | CORRECTION: Always calculate D first. If D is zero, it means there's no unique solution (either infinite solutions or no solution), and Cramer's Rule doesn't apply directly.

MISTAKE: Swapping columns incorrectly when calculating Dx or Dy. For example, putting the constant terms in the wrong column. | CORRECTION: Remember, for Dx, replace the 'x' coefficient column with the constant terms. For Dy, replace the 'y' coefficient column with the constant terms.

MISTAKE: Making calculation errors when finding the determinant (e.g., forgetting the minus sign for the second term in a 2x2 determinant). | CORRECTION: Practice determinant calculations carefully. For a 2x2 matrix |a b| |c d|, the determinant is always (a*d) - (b*c).

Practice Questions

Try It Yourself

QUESTION: Use Cramer's Rule to solve: x + y = 7, x - y = 3 | ANSWER: x = 5, y = 2

QUESTION: Find the value of x using Cramer's Rule for: 2x + 3y = 13, x - y = -1 | ANSWER: x = 2

QUESTION: For the system: 3x - 2y = 10, 5x + 4y = 2, calculate the main determinant D and then find the value of y. | ANSWER: D = 22, y = -2

MCQ

Quick Quiz

For Cramer's Rule to give a unique solution, what must be true about the determinant of the coefficient matrix (D)?

D must be equal to zero

D must be greater than zero

D must not be equal to zero

D must be less than zero

The Correct Answer Is:

For a unique solution using Cramer's Rule, the determinant of the coefficient matrix (D) must not be zero. If D = 0, the system either has no solution or infinitely many solutions.

Real World Connection

In the Real World

In India, companies like Swiggy or Zomato use complex systems of equations to optimize delivery routes and assign riders efficiently. Cramer's Rule, or its advanced versions, helps solve these equations to ensure your food reaches you quickly and at the right price, by finding unique solutions for variables like time, distance, and cost.

Key Vocabulary

Key Terms

DETERMINANT: A special number calculated from a square matrix, used in solving systems of equations. | SYSTEM OF LINEAR EQUATIONS: A set of two or more linear equations with the same variables. | UNIQUE SOLUTION: A single, specific set of values for the variables that satisfies all equations in the system. | COEFFICIENT MATRIX: A matrix formed by the coefficients of the variables in a system of linear equations.

What's Next

What to Learn Next

Now that you understand Cramer's Rule for unique solutions, you can explore what happens when the main determinant is zero – leading to cases of 'no solution' or 'infinite solutions'. This will deepen your understanding of systems of linear equations.