Question

How do you write equation of the line that passes through point $ \left( {4,2} \right) $ and $ \left( {6,6} \right) $

Answer 1

Hint: In order to determine the required equation of line, first find out the value of slope $ m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} $ by considering $ \left( {4,2} \right) $ as\[\left( {{x_1},{y_1}} \right)\]and $ \left( {6,6} \right) $ as\[\left( {{x_2},{y_2}} \right)\].Now put the slope $ m $ and any point in the slope point form $ \left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right) $ and simplify it to obtain the required equation.

Complete step-by-step answer:
We are given two points as $ \left( {4,2} \right) $ and $ \left( {6,6} \right) $ .
In this question we are supposed to find out the equation of line which is passing through the points $ \left( {4,2} \right) $ and $ \left( {6,6} \right) $ .
For this we have to first determine the slope of the line passing through these two points. So, as we know the slope between two points is given by $ m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} $ where\[\left( {{x_1},{y_1}} \right)\]and \[\left( {{x_2},{y_2}} \right)\]are the coordinates of two points.
Considering $ \left( {4,2} \right) $ as\[\left( {{x_1},{y_1}} \right)\]and $ \left( {6,6} \right) $ as\[\left( {{x_2},{y_2}} \right)\], we have the value of slope as
 $
  m = \dfrac{{6 - 2}}{{6 - 4}} \\
  m = \dfrac{4}{2} \\
  m = 2 \;
  $
Thus we get the slope $ m $ equals $ 2 $ .
The Point-Slope Formula of straight line is
 $ \left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right) $ where $ \left( {{x_1},{y_1}} \right) $ is the point on the line .
So, we have the slope of the required line as $ m = 2 $ and also it is passing through the point $ \left( {4,2} \right) $ .We can write the equation of straight line using the point slope form as
\[ \Rightarrow \left( {y - 2} \right) = 2\left( {x - 4} \right)\]
Expanding the bracket on RHS, we get
\[ \Rightarrow y - 2 = 2x - 8\]
combining all the like terms and rewrite the equation into the general equation form as $ y = mx + c $ , we can obtain the above equation as
\[
   \Rightarrow y = 2x - 8 + 2 \\
   \Rightarrow y = 2x - 6 \;
 \]
Therefore, the equation of line passing through the points $ \left( {4,2} \right) $ and $ \left( {6,6} \right) $ is equal to \[y = 2x - 6\].
So, the correct answer is “ \[y = 2x - 6\]”.

Note: 1. The graph of the equation of line\[y = 2x - 6\] is shown below.

You can verify the result as both the points are lying on the straight line.
2.Slope of line perpendicular to the line having slope $ m $ is equal to $ - \dfrac{1}{m} $ .
3.We should have a better knowledge in the topic of geometry to solve this type of question easily. We should know the Point-slope form $ \left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right) $ where $ \left( {{x_1},{y_1}} \right) $ is the point on the line $ m $ as the form and also the Slope-intercept form of line as $ y = mx + c $ where $ m $ is the slope of the line.
4. The general equation for lines parallel to \[y = 2x - 6\]will be \[y = 2x \pm k\]where $ k $ can be any integer.
5. Write the coordinates with proper signs while determining the slope and equation.
6. In the point slope form we have taken $ \left( {4,2} \right) $ . You can also take $ \left( {6,6} \right) $ .