Question

Equation of another line parallel to the line represented by $2x - 6y = 7$ is :

• A. $y = 3x - 7$
• B. $2x = 9 - 6y$
• C. $x - 3y = 7$
• D. $x = \frac{7}{2} - 3y$

Hint:

For a quick visual check, parallel lines have identical or proportional $x$ and $y$ coefficients.
The given line has coefficients $(2x - 6y)$. Dividing this by 2 gives $(x - 3y)$.
Looking at the options, Option (C) has $(x - 3y)$ on the left-hand side.
This matching ratio allows you to instantly recognize the parallel line without calculating slopes!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question is from the chapter "Pair of Linear Equations in Two Variables".
We are given a linear equation $2x - 6y = 7$, which represents a straight line.
We need to identify which of the given options represents a line that is parallel to this line.

Step 2: Key Formula or Approach:
Two lines given by $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are parallel if they have the same slope but different intercepts.
The algebraic condition for parallel lines is:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \] Alternatively, we can find the slope $m$ of the given line using the slope-intercept form ($y = mx + c$) and check which option has the exact same slope $m$.

Step 3: Detailed Explanation:

• Convert the given equation into slope-intercept form to find its slope:
\[ 2x - 6y = 7 \] \[ -6y = -2x + 7 \] Divide by $-6$:
\[ y = \frac{-2}{-6}x + \frac{7}{-6} \] \[ y = \frac{1}{3}x - \frac{7}{6} \] The slope of the given line is $m = \frac{1}{3}$.

• Now, convert each of the option equations into the slope-intercept form ($y = mx + c$) to find their slopes:

• Option (A): $y = 3x - 7$
The slope is $m = 3$. This is not equal to $\frac{1}{3}$. (Not parallel)

• Option (B): $2x = 9 - 6y \implies 6y = -2x + 9 \implies y = -\frac{1}{3}x + \frac{3}{2}$
The slope is $m = -\frac{1}{3}$. This is not equal to $\frac{1}{3}$. (Not parallel)

• Option (C): $x - 3y = 7 \implies -3y = -x + 7 \implies y = \frac{1}{3}x - \frac{7}{3}$
The slope is $m = \frac{1}{3}$. This is equal to the slope of our given line, and the y-intercept ($-\frac{7}{3}$) is different from the original y-intercept ($-\frac{7}{6}$). Therefore, these lines are parallel.

• Option (D): $x = \frac{7}{2} - 3y \implies 3y = -x + \frac{7}{2} \implies y = -\frac{1}{3}x + \frac{7}{6}$
The slope is $m = -\frac{1}{3}$. This is not equal to $\frac{1}{3}$. (Not parallel)


Step 4: Final Answer:
Only the line in Option (C) has the same slope as the given line, indicating that they are parallel lines.