What is Cramer's Rule for Two Variables?

S6-SA1-0187

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

Cramer's Rule for two variables is a clever method to solve a system of two linear equations without using substitution or elimination. It uses something called 'determinants' to find the values of the variables quickly. Think of it as a shortcut to find 'x' and 'y' when you have two equations like ax + by = c.

Simple Example

Quick Example

Imagine you buy 2 samosas and 1 chai for Rs 40. Then, your friend buys 3 samosas and 2 chai for Rs 70. You want to find the price of one samosa (x) and one chai (y). Cramer's Rule can quickly tell you 'x' and 'y' without guessing or doing long calculations.

Worked Example

Step-by-Step

Let's solve:
Equation 1: 2x + 3y = 12
Equation 2: 3x + y = 11

Step 1: Write down the coefficients in a matrix form for the main determinant (D). D = (2*1) - (3*3) = 2 - 9 = -7.
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Step 2: To find Dx, replace the 'x' coefficients with the constant terms. Dx = (12*1) - (3*11) = 12 - 33 = -21.
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Step 3: To find Dy, replace the 'y' coefficients with the constant terms. Dy = (2*11) - (12*3) = 22 - 36 = -14.
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Step 4: Calculate x = Dx / D. So, x = -21 / -7 = 3.
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Step 5: Calculate y = Dy / D. So, y = -14 / -7 = 2.
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Answer: So, x = 3 and y = 2.

Why It Matters

Cramer's Rule is super useful in fields like AI/ML to solve complex problems involving many variables and in Engineering to design structures or circuits. Knowing this rule can open doors to exciting careers in data science, robotics, or even space technology at ISRO, where quick and accurate calculations are key.

Common Mistakes

MISTAKE: Swapping the order of multiplication in a determinant (e.g., (b*c) - (a*d) instead of (a*d) - (b*c)). | CORRECTION: Always remember it's (top-left * bottom-right) - (top-right * bottom-left).

MISTAKE: Forgetting to replace the correct column (x or y coefficients) with the constant terms when calculating Dx or Dy. | CORRECTION: For Dx, replace the first column (x-coefficients). For Dy, replace the second column (y-coefficients).

MISTAKE: Making calculation errors, especially with negative numbers. | CORRECTION: Double-check your multiplication and subtraction, especially when dealing with minus signs. A small error can change the entire answer.

Practice Questions

Try It Yourself

QUESTION: Solve for x and y using Cramer's Rule:
x + y = 7
2x - y = 2
| ANSWER: x = 3, y = 4

QUESTION: Find the value of 'x' using Cramer's Rule for the system:
3x - 2y = 10
4x + 5y = 6
| ANSWER: x = 4

QUESTION: A shop sells two types of mobile covers. Type A costs Rs 150 and Type B costs Rs 200. If 5 covers of Type A and 3 covers of Type B cost Rs 1350, and 2 covers of Type A and 4 covers of Type B cost Rs 1100, find the individual cost of each type of cover using Cramer's Rule. (Hint: Form two equations first)
| ANSWER: Cost of Type A = Rs 100, Cost of Type B = Rs 250

MCQ

Quick Quiz

Which determinant is used in the denominator when applying Cramer's Rule?

Determinant of the x-coefficients and constant terms

Determinant of the y-coefficients and constant terms

Determinant of the coefficients of the variables

Determinant of the constant terms

The Correct Answer Is:

The denominator in Cramer's Rule is always the determinant of the main coefficient matrix (D), which contains the coefficients of both x and y. Options A and B are for Dx and Dy, and option D is incorrect.

Real World Connection

In the Real World

Imagine a logistics company like Delhivery or Ecom Express that needs to figure out the fastest routes for its delivery vans. They might use systems of equations to optimize fuel consumption and delivery times. Cramer's Rule, or its extended versions, helps computers quickly solve these equations to ensure your packages arrive on time, every time.

Key Vocabulary

Key Terms

DETERMINANT: A special number calculated from a square arrangement of numbers. | COEFFICIENT: The number multiplying a variable (e.g., '2' in 2x). | LINEAR EQUATION: An equation whose graph is a straight line. | SYSTEM OF EQUATIONS: A set of two or more equations with the same variables. | MATRIX: A rectangular array of numbers arranged in rows and columns.

What's Next

What to Learn Next

Great job understanding Cramer's Rule for two variables! Next, you can explore 'Cramer's Rule for Three Variables' to solve even bigger problems. This will help you tackle more complex situations in algebra and prepare you for higher studies in science and engineering.