What is the cross multiplication method class 10?

The cross multiplication method, taught in Class 10 mathematics, is a technique used to solve pairs of linear equations in two variables. According to the provided reference, it is considered a simple and accurate method for finding the values of the variables.

Understanding the Cross Multiplication Method

Here's a breakdown of how the cross multiplication method works:

General Form of Linear Equations

First, consider a pair of linear equations in the general form:

• a₁x + b₁y + c₁ = 0
• a₂x + b₂y + c₂ = 0

Where:

• x and y are the variables.
• a₁, a₂, b₁, b₂, c₁, and c₂ are constants.

The Formula

The cross multiplication method provides a direct formula to find the values of x and y:

x / (b₁c₂ - b₂c₁) = y / (c₁a₂ - c₂a₁) = 1 / (a₁b₂ - a₂b₁)

Solving for x and y

From the above equation, we can find x and y as follows:

• To find x: x = (b₁c₂ - b₂c₁) / (a₁b₂ - a₂b₁)
• To find y: y = (c₁a₂ - c₂a₁) / (a₁b₂ - a₂b₁)

Steps to Apply the Method

1. Write the equations in the standard form: Ensure your equations are in the form a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.
2. Identify the coefficients: Determine the values of a₁, a₂, b₁, b₂, c₁, and c₂.
3. Apply the formula: Substitute the values into the cross multiplication formula.
4. Solve for x and y: Calculate the values of x and y using the derived formulas.

Example

Let's solve the following system of equations using cross-multiplication:

• 2x + 3y - 8 = 0
• 4x + y - 2 = 0

1. Standard form: The equations are already in standard form.

2. Coefficients:

   • a₁ = 2, b₁ = 3, c₁ = -8
   • a₂ = 4, b₂ = 1, c₂ = -2

3. Apply the Formula:
   x / (3(-2) - 1(-8)) = y / ((-8)4 - (-2)2) = 1 / (21 - 43)

   x / (-6 + 8) = y / (-32 + 4) = 1 / (2 - 12)

   x / 2 = y / -28 = 1 / -10

4. Solve for x and y:

   • x = (2) / (-10) = -1/5
   • y = (-28) / (-10) = 14/5

Therefore, the solution is x = -1/5 and y = 14/5.

When to Use Cross Multiplication

As the reference states, cross multiplication is best suited when dealing with pairs of linear equations in two variables. It provides a direct method to find the solution, avoiding the need for substitution or elimination in some cases.