Question

Find the slope of the line which passes through the points \[P\left( {3,2} \right)\] and \[Q\left( {5,6} \right)\].

Answer 1

Hint: The given question deals with the concept of finding the slope of a line in terms of the coordinates of two points on the line. In order to solve the given question, we will use the formula for finding slope which is \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\] in which \[\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)\] are the two points through which the line passes. With the help of this formula, we will determine the value of the slope.

Complete step by step solution:
Given two points through which the line passes are, P \[\left( {3,2} \right)\]and Q \[\left( {5,6} \right)\]
We know that the slope of a non-vertical line that passes through the points \[A\left( {{x_1},{y_1}} \right)and B\left( {{x_2},{y_2}} \right)\]is given by \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
Here, from point P we get,\[{x_1} = 3\] and \[{y_1} = 2\]
And from the point Q we get, \[{x_2} = 5\]and \[{y_2} = 6\]
Therefore, putting these values in the formula
We get,
\[ \Rightarrow m = \dfrac{{6 - 2}}{{5 - 3}}\]
Thus, \[m = \dfrac{4}{2} = 2\]

Hence, the value of the required slope is $2$.

Note: Alternative solution for the given question is as follows:
 It is important to note here that the equation of a line L that passes through the points \[\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)\] is given by \[y - {y_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}(x - {x_2})\]
Thus, we can write the equation of the line that passes through the given points \[P\left( {3,2} \right)\]and \[Q\left( {5,6} \right)\] with the help for this formula.
Here, \[{x_1} = 3\],\[{y_1} = 2\] and \[{x_2} = 5\], \[{y_2} = 6\].
Therefore, putting these values in the equation of line stated above, we get,
\[ \Rightarrow y - 2 = \dfrac{{6 - 2}}{{5 - 3}}(x - 5)\]
\[ \Rightarrow y - 2 = \dfrac{4}{2}(x - 5)\]
Thus,
\[ \Rightarrow y - 2 = 2(x - 5)\]
\[ \Rightarrow y - 2 = 2x - 10\]
Hence, the equation of the line is
\[ \Rightarrow y = 2x - 8 - - - - - (1)\]
We know that the standard equation of a line is \[y = mx + c\].
Therefore comparing it with equation (1)
We get, \[m = 2\]
Which is the required slope.