Question

How do you write the equation of a line with point \[\left( 4,2 \right)\] and slope \[\dfrac{1}{2}\]?

Answer 1

Hint: In order to find the solution of the given question that is to write the equation of a line with point \[\left( 4,2 \right)\] and slope \[\dfrac{1}{2}\] use the standard equation of point-slope formula \[\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right)\] where \[m\] is the slope of the line and \[\left( {{x}_{1}},{{y}_{1}} \right)\] is the point that passes through the line. As here, we have the value of slope and point passing through the line then we can put these values in the standard formula to find the required equation.

Complete step-by-step solution:
According to the question,
The slope in the question is \[\dfrac{1}{2}\] and the point that passes through the line \[\left( 4,2 \right)\].
We know that the standard equation of point-slope formula \[\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right)\] where \[m\] is the slope of the line and \[\left( {{x}_{1}},{{y}_{1}} \right)\] is the point that passes through the line.
Now, substituting the given values in the above formula we will have:
\[\Rightarrow \left( y-2 \right)=\dfrac{1}{2}\left( x-4 \right)\]
After solving the bracket, we get:
\[\Rightarrow y-2=\dfrac{x}{2}-2\]
As we can see, \[-2\] is there on both sides of the above equation and hence gets cancelled out. So, we are left with:
\[\Rightarrow y=\dfrac{x}{2}\]
Therefore, the equation of line with slope as \[\dfrac{1}{2}\] and the point that passes through the line \[\left( 4,2 \right)\] is \[y=\dfrac{x}{2}\].

Note: Students can go wrong by using the wrong point-slope formula that is they use this \[\left( {{y}_{1}}-y \right)=m\left( {{x}_{1}}-x \right)\] which is completely wrong and leads to the wrong answer. We must know which formula we need to use, and we should apply it right. The correct point-slope formula is \[\left( y-{{y}_{1}} \right)=m\left( x-{{x}_{1}} \right)\] where \[m\] is the slope of the line and \[\left( {{x}_{1}},{{y}_{1}} \right)\] is the point that passes through the line.