Step 1: Understand the Question:
This question is from Pair of Linear Equations in Two Variables.
We are given the line $2.5x - 2y = 3$ and need to pick, from four options, the equation that represents the exact same line (a coincident line).
Step 2: Write down the coincidence rule and the coefficients of the given line.
Two lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are coincident only when all three ratios match:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]
Write the given line in standard form:
\[ 2.5x - 2y - 3 = 0 \]
So $a_1 = 2.5$, $b_1 = -2$, $c_1 = -3$.
Step 3: Test options (A), (B) and (C) and show they fail.
Option (A), $5x - 4y - 3 = 0$: here $a_2=5, b_2=-4, c_2=-3$.
\[ \frac{a_1}{a_2} = \frac{2.5}{5} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{-2}{-4} = \frac{1}{2}, \quad \frac{c_1}{c_2} = \frac{-3}{-3} = 1 \]
The first two ratios equal $\frac{1}{2}$ but the third equals 1, so this fails the coincidence test (it is only a parallel line).
Option (B), $5x - 4y + 6 = 0$: here $c_2 = 6$, so $\frac{c_1}{c_2} = \frac{-3}{6} = -\frac{1}{2}$, which does not equal $\frac{1}{2}$. This also fails.
Option (C), $15x - 12y - 3 = 0$: here $a_2=15, b_2=-12, c_2=-3$.
\[ \frac{a_1}{a_2} = \frac{2.5}{15} = \frac{1}{6}, \quad \frac{c_1}{c_2} = \frac{-3}{-3} = 1 \]
These do not match either, so this fails too.
Step 4: Test option (D).
Option (D), $5x - 4y - 6 = 0$: here $a_2 = 5$, $b_2 = -4$, $c_2 = -6$.
\[ \frac{a_1}{a_2} = \frac{2.5}{5} = \frac{1}{2}, \quad \frac{b_1}{b_2} = \frac{-2}{-4} = \frac{1}{2}, \quad \frac{c_1}{c_2} = \frac{-3}{-6} = \frac{1}{2} \]
All three ratios equal $\frac{1}{2}$, so this line is coincident with the given line.
Final Answer:
The equation coincident with the given line is $5x - 4y - 6 = 0$, which matches option (D).
\[ \boxed{5x - 4y - 6 = 0} \]