Question

If the straight lines $4x+6y=5$ and $6x+ky=3$ are parallel, then the value of k is equal to

• A. $\frac{-2}{3}$
• B. 8
• C. 9
• D. 10
• E. $\frac{3}{2}$

Hint:

Shortcut Tip: For parallel lines $A_1x + B_1y = C_1$ and $A_2x + B_2y = C_2$, their coefficients are simply proportional: $\frac{A_1}{A_2} = \frac{B_1}{B_2}$. So, $\frac{4}{6} = \frac{6}{k} \implies 4k = 36 \implies k = 9$.

Answer 1

The Correct Option is C

Concept:
Two straight lines are parallel if and only if their slopes are equal ($m_1 = m_2$). For any linear equation in the standard form $Ax + By = C$, the slope of the line can be quickly found using the formula $m = -\frac{A}{B}$.

Step 1: Find the slope of the first line.
The first equation is $4x + 6y = 5$. Using $m = -\frac{A}{B}$, where $A = 4$ and $B = 6$: $$m_1 = -\frac{4}{6}$$ $$m_1 = -\frac{2}{3}$$

Step 2: Find the slope of the second line.
The second equation is $6x + ky = 3$. Using $m = -\frac{A}{B}$, where $A = 6$ and $B = k$: $$m_2 = -\frac{6}{k}$$

Step 3: Equate the two slopes.
Since the problem states the lines are parallel, we set their slopes equal to each other: $$-\frac{2}{3} = -\frac{6}{k}$$

Step 4: Cross-multiply to solve for k.
Cancel the negative signs on both sides and cross-multiply: $$2 \times k = 3 \times 6$$ $$2k = 18$$

Step 5: Calculate the final value.
Divide by 2 to isolate $k$: $$k = 9$$ Hence the correct answer is (C) 9.