Question

If the line $2x + 3y = 5$ and $y = mx + c$ be parallel, then
A. $m = \dfrac{2}{3},c = 5$
B. $m = - \dfrac{2}{3},c = 5$
C. $m = - \dfrac{2}{3},c = {\text{any real number}}$
D. None of these

Answer 1

Hint: We are given an equation of two straight lines $2x + 3y = 5$ and $y = mx + c$ that are parallel to each other. We must know the slope intercept form of the straight line. So, we know that the slope of parallel lines is equal to each other. We will apply the condition for two lines to be parallel on the equations of lines provided to us.

Complete step by step answer:
So, we have, $2x + 3y = 5$ and $y = mx + c$. We first convert the equation $2x + 3y = 5$ into a slope intercept form of a straight line. We know that the slope intercept from a straight line is $y = mx + k$. So, isolating the variable y to resemble the slope intercept form of a straight line, we get,
$ \Rightarrow 3y = 5 - 2x$
$ \Rightarrow y = \left( {\dfrac{{5 - 2x}}{3}} \right)$
Simplifying the equation, we get,
$ \Rightarrow y = - \dfrac{{2x}}{3} + \dfrac{5}{3}$

Now, comparing the equation with slope and intercept form, we get the value of slope as $m = - \dfrac{2}{3}$ and y intercept as $k = \dfrac{5}{3}$. Now, since the equation $y = mx + c$ is parallel to the former line. So, the slope of both the lines is equal. Hence, we get the slope of the line $y = mx + c$ is $m = - \dfrac{2}{3}$. Now, for the lines to be parallel, the value of slope should be the same while y intercept can take any value. So, we get, $m = - \dfrac{2}{3}$ and c to be any real number.

Hence, option C is the correct answer.

Note: We know the equation of a line passing through a point and having a slope ‘m’ and with ‘y’ intercept as ‘c’ is given by $y = mx + c$. ‘y’ intercept is defined as a line or a curve crosses the y-axis of a graph. In other words the value of ‘y’ at ‘x’ is equal to zero. Hence, the y intercept of a line can also be found by putting the value of x as zero. Slope of a line is the inclination of the straight line with positive x-axis.