Question

How to determine the equation of the line parallel to \[3x - 2y + 4 = 0\] and passing through \[(1,6)\]?

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 point-slope formula to write the equation of the line passing through the given point and having the slope of the given equation of line.
* General equation of line is \[y = mx + c\]
* Point-slope formula: Equation of line passing through the point \[({x_1},{y_1})\] and having slope ‘m’ is given by \[y - {y_1} = m(x - {x_1})\].

Complete step-by-step solution:
We are given the equation of line as \[3x - 2y + 4 = 0\]
We can shift 2y to right hand side of the equation
\[ \Rightarrow 3x + 4 = 2y\]
Divide both sides of the equation by 2
\[ \Rightarrow \dfrac{{3x + 4}}{2} = \dfrac{{2y}}{2}\]
Cancel same factors from both numerator and denominator
\[ \Rightarrow \dfrac{{3x}}{2} + 2 = y\]
So, we have the equation as \[y = \dfrac{3}{2}x + 2\].
Compare the equation of line with general equation of line \[y = mx + c\]
We get \[m = \dfrac{3}{2}\].............… (1)
We know that the slope of parallel lines is equal. So, the slope of the line required is also \[\dfrac{3}{2}\]
Now use the point-slope formula with slope \[m = \dfrac{3}{2}\] and point \[(1,6)\] to form an equation of parallel lines.
We know equation of line passing through the point \[({x_1},{y_1})\] and having slope ‘m’ is given by\[y - {y_1} = m(x - {x_1})\].
So, equation of line with slope \[m = \dfrac{3}{2}\] and point \[(1,6)\] passing through it is
\[ \Rightarrow y - 6 = \dfrac{3}{2}(x - 1)\]
Cross multiply 2 from denominator of RHS to LHS
\[ \Rightarrow 2y - 12 = 3x - 3\]
Bring all terms to LHS of the equation
\[ \Rightarrow 2y - 12 - 3x + 3 = 0\]
\[ \Rightarrow 2y - 3x - 9 = 0\]
Multiply both sides of the equation by -1
\[ \Rightarrow 3x - 2y + 9 = 0\]

\[\therefore \]The equation of the line parallel to \[3x - 2y + 4 = 0\] and passing through \[(1,6)\] is \[3x - 2y + 9 = 0\].

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 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.