Question

A line has the equation \[2y = - 3x + 1\] . How do you find an equation of a line parallel to this line that has a y intercept of -2?

Answer 1

Hint: We first find the value of slope by comparing the given equation of line to the general equation of line. Use the concept of parallel lines having the same slope and write slope of the parallel line equal to slope of given line. Use slope intercept form to write the equation of the line having y-intercept and having the slope of the given equation of line.
* General equation of the line is \[y = mx + c\] , where ‘m’ is the slope of the line and ‘c’ is the value of the y-intercept.

Complete step by step solution:
We are given the equation of line as \[2y = - 3x + 1\]
Divide both sides of the equation by 2
\[ \Rightarrow \dfrac{{2y}}{2} = \dfrac{{ - 3x + 1}}{2}\]
Cancel same factors from both numerator and denominator
\[ \Rightarrow y = \dfrac{{ - 3}}{2}x + \dfrac{1}{2}\]
So, we have the equation as \[y = \dfrac{{ - 3}}{2}x + \dfrac{1}{2}\].
Compare the equation of line with general equation of line \[y = mx + c\]
We get \[m = \dfrac{{ - 3}}{2}\] … (1)
And the value of y-intercept is \[\dfrac{1}{2}\]
We know that the slope of parallel lines is equal. So, the slope of the line required is also \[\dfrac{3}{2}\]
Now use slope intercept formula with slope \[m = \dfrac{{ - 3}}{2}\]and y-intercept as -2
So, equation of line with slope \[m = \dfrac{3}{2}\]and y-intercept -2 is:
\[ \Rightarrow y = \dfrac{{ - 3}}{2}x + ( - 2)\]
Multiply sign outside the bracket to the sign inside the bracket
\[ \Rightarrow y = \dfrac{{ - 3}}{2}x - 2\]
Take LCM in right hand side of the equation
\[ \Rightarrow y = \dfrac{{ - 3x - 4}}{2}\]
Cross multiply values from denominator of right hand side to left hand side of the equation
\[ \Rightarrow 2y = - 3x - 4\]
Take all values to left hand side of the equation
\[ \Rightarrow 3x + 2y + 4 = 0\]
\[\therefore \] The equation of the line parallel to \[2y = - 3x + 1\] and having y-intercept -2 is \[3x + 2y + 4 = 0\].

Note: Many students make the mistake of calculating the slope of the equation of line given wrong as they bring y to one side but they don’t focus on making the coefficient of y as 1 which is wrong. Keep in mind, to compare the equation to the general equation of line it must be similar to that form. Also, write the value of y-intercept as given in the question for the new line, i.e. substitute the value of ‘c’ according to the question.