Question

Find the slope of line passing through \[( - 2,5)\]&\[(6,4)\].

Answer 1

Hint: The slope of a line in the plane containing the \[x\] and \[y\] axes in generally represented by the letter m, and is defined by as the change in the \[y\]coordinates divided by the corresponding change in the \[x\]coordinate, between two distinct point on the line.
This is described by equation:
\[m = \dfrac{{\mathop y\nolimits_2 - \mathop y\nolimits_1 }}{{\mathop x\nolimits_2 - \mathop x\nolimits_1 }}\]
Where points \[(\mathop x\nolimits_1 ,\mathop y\nolimits_1 )\]and \[(\mathop x\nolimits_2 ,\mathop y\nolimits_2 )\] the change in \[x\] from one to the other is \[\mathop x\nolimits_2 - \mathop x\nolimits_1 \](run). While the change in \[y\] is \[\mathop y\nolimits_2 - \mathop y\nolimits_1 \] (rise) substituting both quantities into the above equation generates the formula.

Complete step-by- step solution:
Given, A line run through two points \[P = ( - 2,5)\]\[Q = (6,4)\]
Here in \[P = ( - 2,5)\] represents \[{x_1} , {y_1}\] where \[{x_1} = - 2,{y_1} = 5\]
And also \[Q = (6,4)\] represents \[{x_2} , {y_2}\] where \[{x_2} = 6,{y_2} = 4\]
We know that the slope of line is given by
\[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\]
Substitute the values \[{x_1} = - 2,{x_2} = 6\,and\,{y_1} = 5,{y_2} = 4\] we get,
\[m(slope) = \dfrac{{4 - 5}}{{6 - ( - 2)}}\]
\[m = \dfrac{{ - 1}}{{6 + 2}}\]
\[m = \dfrac{{ - 1}}{8}\]
Required slope of line \[m = \dfrac{{ - 1}}{8}\]

Note: A line is a one-dimensional figure, which has length but no width. A line made of a set of points which is extended in opposite directions infinitely. It is determined by two points in a two-dimensional plane. The two points which lie on the same line are said to be collinear points.
In geometry, there are different types of lines such as horizontal and vertical lines, parallel and perpendicular lines. These lines play an important role in the construction of different types of polygons.
For example, a square is made by four lines of the same lengths, whereas a triangle is made by joining three lines end to end. The above formula fails for a vertical line, parallel to\[y - axis\], where the slope can be taken as infinite. Since the slope is negative, the direction of the line is decreasing.