What is Cross-Multiplication Method? | Simple Explanation for Class 10

S6-SA1-0009

What is the Cross-Multiplication Method for Linear Equations?

Grade Level:

Class 10

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

Definition

What is it?

The Cross-Multiplication Method is a simple algebraic technique used to solve a system of two linear equations with two variables. It helps you find the values of 'x' and 'y' by cross-multiplying the coefficients of the variables and the constant terms.

Simple Example

Quick Example

Imagine you go to a snack shop. You buy 2 samosas and 3 jalebis for Rs 40 on one day. The next day, you buy 3 samosas and 2 jalebis for Rs 50. Using cross-multiplication, you can figure out the exact price of one samosa and one jalebi!

Worked Example

Step-by-Step

Let's solve the system of equations:
1) 2x + 3y = 11
2) 3x + 2y = 9

---Step 1: Rewrite equations in standard form (ax + by + c = 0).
2x + 3y - 11 = 0
3x + 2y - 9 = 0

---Step 2: Identify coefficients (a1, b1, c1, a2, b2, c2).
a1 = 2, b1 = 3, c1 = -11
a2 = 3, b2 = 2, c2 = -9

---Step 3: Apply the cross-multiplication formula:
x / (b1c2 - b2c1) = y / (c1a2 - c2a1) = 1 / (a1b2 - a2b1)

---Step 4: Substitute the values:
x / ((3)(-9) - (2)(-11)) = y / ((-11)(3) - (-9)(2)) = 1 / ((2)(2) - (3)(3))

---Step 5: Calculate the denominators:
x / (-27 - (-22)) = y / (-33 - (-18)) = 1 / (4 - 9)
x / (-27 + 22) = y / (-33 + 18) = 1 / (-5)
x / (-5) = y / (-15) = 1 / (-5)

---Step 6: Solve for x and y.
x / (-5) = 1 / (-5) => x = (-5) / (-5) => x = 1
y / (-15) = 1 / (-5) => y = (-15) / (-5) => y = 3

---Answer: So, x = 1 and y = 3.

Why It Matters

This method is super useful in fields like Engineering and Physics to quickly solve systems of equations that describe how things move or interact. Scientists in Biotechnology might use it to calculate chemical concentrations, and even AI/ML engineers use similar logic in algorithms. It's a foundational skill for problem-solving in many advanced careers.

Common Mistakes

MISTAKE: Forgetting to rewrite equations in the standard form ax + by + c = 0, especially for the constant term 'c'. | CORRECTION: Always move the constant term to the left side of the equation so that the right side is 0. For example, 2x + 3y = 11 should become 2x + 3y - 11 = 0.

MISTAKE: Mixing up the order of coefficients in the formula, like using (b2c1 - b1c2) instead of (b1c2 - b2c1). | CORRECTION: Memorize the correct cyclic order of coefficients (b-c, c-a, a-b) or write them down clearly before substituting. A good trick is to write b1 b2, c1 c2, a1 a2, b1 b2 and cross-multiply diagonally.

MISTAKE: Incorrectly handling negative signs during multiplication. For example, -(-11) becoming -11 instead of +11. | CORRECTION: Be extra careful with signs. Remember that multiplying two negative numbers gives a positive result, and subtracting a negative number is the same as adding a positive number.

Practice Questions

Try It Yourself

QUESTION: Solve for x and y: 3x + 4y = 10 and 2x - 2y = 2 | ANSWER: x = 2, y = 1

QUESTION: Find the values of 'p' and 'q' if p + 2q = 5 and 3p - q = 1 | ANSWER: p = 1, q = 2

QUESTION: A shop sells two types of sweets, ladoo (L) and barfi (B). If 4L + 5B costs Rs 140, and 6L + 3B costs Rs 150, find the cost of one ladoo and one barfi. | ANSWER: Cost of 1 Ladoo = Rs 20, Cost of 1 Barfi = Rs 12

MCQ

Quick Quiz

Which of the following is the correct denominator for 'x' in the cross-multiplication formula for a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0?

a1b2 - a2b1

b1c2 - b2c1

c1a2 - c2a1

a1c2 - a2c1

The Correct Answer Is:

The correct denominator for 'x' in the cross-multiplication method is b1c2 - b2c1. This part of the formula comes from cross-multiplying the 'y' coefficients and constant terms.

Real World Connection

In the Real World

This method is like how a delivery app's system calculates the best routes. Imagine a delivery agent (like for Swiggy or Zomato) has two different routes with different traffic conditions and distances. By setting up equations for time and distance, the system can quickly solve for unknown variables (like average speed needed) using methods similar to cross-multiplication to optimize delivery times and fuel usage.

Key Vocabulary

Key Terms

LINEAR EQUATION: An equation whose graph is a straight line, involving variables raised to the power of 1. | COEFFICIENT: A number multiplied by a variable in an algebraic term. | CONSTANT TERM: A term in an algebraic expression that has a fixed value and does not contain any variables. | SYSTEM OF EQUATIONS: A set of two or more equations that contain two or more variables. | VARIABLE: A symbol, usually a letter, representing an unknown value.

What's Next

What to Learn Next

Great job mastering cross-multiplication! Next, you should explore the 'Elimination Method' and 'Substitution Method' for solving linear equations. Understanding all these methods will give you a complete toolkit for tackling any system of linear equations, preparing you for more complex algebra in higher grades.