Question

Derive the equation of the straight line passing through the point \[\left( {{x_1},{y_1}} \right)\] and having the slope \['m'\].

Answer 1

Hint: Here, we have to derive the equation of a straight line. This can be derived using the one point and one slope formula. This derived equation can also be used to find the equation of the straight line. The slope of a line is a measure of its steepness.

Formula used:
 Equation of a straight line is given by \[y = mx + c\]
where \[m\] is the slope of the straight line and \[c\] is the \[y\]- intercept.

Complete step-by-step answer:
We are given that a straight line is passing through the point \[\left( {{x_1},{y_1}} \right)\] and having the slope \['m'\].
Equation of a straight line is given by \[y = mx + c\].
where \[m\] is the slope of the straight line \[m = {\rm{slope c}}\]= Constant.
The Point \[\left( {{x_1},{y_1}} \right)\] satisfies \[y = mx + c\] as it passes through the points, we get
\[ \Rightarrow {y_1} = m{x_1} + c\]
Rewriting the equation, we have
\[ \Rightarrow c = {y_1} - m{x_1}\] ……………………….\[\left( 1 \right)\]
Substituting equation \[\left( 1 \right)\] in equation \[y = mx + c\], we get
\[ \Rightarrow y = mx + {y_1} - m{x_1}\]
Rewriting the equation, we get
\[ \Rightarrow y - {y_1} = m\left( {x - {x_1}} \right)\]
This is the required equation of a straight line.

Note: We can derive the same equation using the slope with two point formula. We can consider two points say \[\left( {x,y} \right)\] and \[\left( {{x_1},{y_1}} \right)\] and having the slope \['m'\]. By using the two point formula, we derive the equation of a straight line. Equation of a line can be of general form, standard form, point – slope form, slope – intercept form, intercept form, two point form. The equation of a line can also be used to find the slope of a straight line. The two properties of straight lines in Euclidean geometry are that they have only one dimension, length, and they extend in two directions forever.