Question

The value of k for which the pair of linear equations \(kx - 3y = 5\), \(4x - 6y = 10\) has infinitely many solutions, is :

• A. 4
• B. 3
• C. 2
• D. 1

Hint:

Always convert the linear equations to the standard form \(ax + by + c = 0\) before extracting coefficients to prevent sign-related errors, particularly for the constant term.
Additionally, verify that the ratio of constants \(c_1/c_2\) is equal to the other ratios to guarantee infinitely many solutions rather than no solution (parallel lines).

Answer 1

The Correct Option is C

Step 1: Understand what is being asked.
We are given $kx-3y=5$ and $4x-6y=10$, and we need the value of $k$ for which the pair has infinitely many solutions. Instead of applying the general ratio condition, let us simplify the second equation and compare it directly with the first.

Step 2: Simplify the second equation.
Every term in $4x-6y=10$ is divisible by 2. Divide throughout by 2:
\[ \frac{4x}{2} - \frac{6y}{2} = \frac{10}{2} \]
\[ 2x - 3y = 5 \]
Step 3: Compare with the first equation.
The first equation is $kx - 3y = 5$, and we have just simplified the second equation to $2x-3y=5$.
For two equations in the same two variables to represent the same line (which is what infinitely many solutions means for a pair of linear equations), every term must match once both equations are written in their simplest, reduced form.
Here the $y$-term and the constant term already match ($-3y$ and $5$ on both sides), so the coefficient of $x$ must also match for the lines to coincide:
\[ k = 2 \]
Step 4: Check the answer.
With $k=2$, the first equation becomes $2x-3y=5$, which is exactly the reduced form of the second equation. So the two equations describe the same line, giving infinitely many common solutions.

Final Answer:
The value of $k$ is 2, matching option (C).
\[ \boxed{k = 2} \]