Question

Equation of a line coincident with 2.5x - 2y = 3 is :

• A. 5x - 4y = 3
• B. 5x - 4y + 6 = 0
• C. 15x - 12y - 3 = 0
• D. 5x - 4y - 6 = 0

Hint:

To quickly find a coincident line, eliminate decimal coefficients by multiplying the entire equation by a suitable integer.
Here, multiplying \( 2.5x - 2y = 3 \) by 2 immediately yields \( 5x - 4y = 6 \), which is equivalent to \( 5x - 4y - 6 = 0 \).
This allows you to bypass verifying the ratio condition for every single option.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The topic of this question is Pair of Linear Equations in Two Variables.
We are given a linear equation in two variables: \( 2.5x - 2y = 3 \).
We need to find another equation from the given options that represents a coincident line with the given line.

Step 2: Key Formula or Approach:
For two linear equations \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) to represent coincident lines, the ratio of their coefficients must satisfy the following condition:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
This means that one equation is simply a constant non-zero multiple of the other equation.

Step 3: Detailed Explanation:
1. First, let us rewrite the given linear equation in standard form \( ax + by + c = 0 \):
\[ 2.5x - 2y - 3 = 0 \]
Here, we identify the coefficients:
- \( a_1 = 2.5 \)
- \( b_1 = -2 \)
- \( c_1 = -3 \)
2. Let us look at the coefficients of the options. Many options have \( 5x - 4y \) as their leading terms.
Let us multiply our standard form equation by a constant factor of 2 to see if we can match the coefficients:
\[ 2 \times (2.5x - 2y - 3) = 2 \times 0 \]
\[ 5x - 4y - 6 = 0 \]
3. Let us identify the coefficients of this new equation:
- \( a_2 = 5 \)
- \( b_2 = -4 \)
- \( c_2 = -6 \)
4. Let us verify the ratio condition to confirm they are coincident:
\[ \frac{a_1}{a_2} = \frac{2.5}{5} = \frac{1}{2} \]
\[ \frac{b_1}{b_2} = \frac{-2}{-4} = \frac{1}{2} \]
\[ \frac{c_1}{c_2} = \frac{-3}{-6} = \frac{1}{2} \]
5. Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = \frac{1}{2} \), the two lines are indeed coincident.
The equation \( 5x - 4y - 6 = 0 \) matches option (D).

Step 4: Final Answer:
The equation of the line coincident with the given line is \(5x - 4y - 6 = 0\), which is option (D).