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

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

Explanation

Solution

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=y2−y1x2−x1m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} in which (x1,y1),(x2,y2)\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 (3,2)\left( {3,2} \right)and Q (5,6)\left( {5,6} \right)
We know that the slope of a non-vertical line that passes through the points A(x1,y1)andB(x2,y2)A\left( {{x_1},{y_1}} \right)and B\left( {{x_2},{y_2}} \right)is given by m=y2−y1x2−x1m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}
Here, from point P we get,x1=3{x_1} = 3 and y1=2{y_1} = 2
And from the point Q we get, x2=5{x_2} = 5and y2=6{y_2} = 6
Therefore, putting these values in the formula
We get,
⇒m=6−25−3\Rightarrow m = \dfrac{{6 - 2}}{{5 - 3}}
Thus, m=42=2m = \dfrac{4}{2} = 2

Hence, the value of the required slope is 22.

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 (x1,y1),(x2,y2)\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right) is given by y−y1=y2−y1x2−x1(x−x2)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(3,2)P\left( {3,2} \right)and Q(5,6)Q\left( {5,6} \right) with the help for this formula.
Here, x1=3{x_1} = 3,y1=2{y_1} = 2 and x2=5{x_2} = 5, y2=6{y_2} = 6.
Therefore, putting these values in the equation of line stated above, we get,
⇒y−2=6−25−3(x−5)\Rightarrow y - 2 = \dfrac{{6 - 2}}{{5 - 3}}(x - 5)
⇒y−2=42(x−5)\Rightarrow y - 2 = \dfrac{4}{2}(x - 5)
Thus,
⇒y−2=2(x−5)\Rightarrow y - 2 = 2(x - 5)
⇒y−2=2x−10\Rightarrow y - 2 = 2x - 10
Hence, the equation of the line is
⇒y=2x−8−−−−−(1)\Rightarrow y = 2x - 8 - - - - - (1)
We know that the standard equation of a line is y=mx+cy = mx + c.
Therefore comparing it with equation (1)
We get, m=2m = 2
Which is the required slope.