Slope Intercept Form of a Straight Line Equation

Go back to  'Straight Lines'

 \(\textbf{Art 4  :} \qquad\boxed{{\text{Equation(s) representing a straight line}}}\)

The last three articles dealt with the preliminaries of co-ordinate geometry and certain elementary formulae which find widespread use. With this article, we start the discussion of the geometry of straight lines in detail.

On the co-ordinate plane, the simplest case for a straight line would be one in which the line is parallel to one of the co-ordinate axes.

As described in the figure above, the equation of such a line is \(y={{y}_{0}}\,\,\text{or}\,\,x={{x}_{0}}\) accordingly as the line is parallel to the x-axis or the y-axis respectively.

These are special cases of lines; we want to find the equation of any arbitrary line in general. Visualise any such line in your mind. To completely specify such a line, you would need two quantities: the inclination of the line (or its slope or the angle it makes with say, the x-axis) and the placement of the line (i.e. where the line passes through with reference to the axes: we can specify the placement of the line by specifying the point on the y-axis through which the line passes, or in other words, by specifying the y-intercept.)

It should be obvious to you that any line can be determined uniquely using these two parameters.

We now find out the equation of this straight line, assuming that we know \(\theta \,\,\text{and}\,\,c.\)  In other words, we intend to find out the relation that the co-ordinates (x, y) of any arbitrary point on the line must satisfy. The determination of this equation is straightforward:

As described in the figure above, we have in \(\Delta \,APB,\)

\[\tan \theta =\frac{PB}{AB}\]

\(\tan \theta \) is a measure of the inclination of the line (its steepness). \(\tan \theta \)  is therefore termed the slope of the line and is denoted by m. Thus, \(m = \tan \theta \) . Also, notice that \(PB=\left( y-c \right)\,\,\text{and}\,\,AB=x.\) Therefore,

\[\begin{align} & \qquad\qquad m = \frac{{y - c}}{x}\\& \Rightarrow  \qquad \boxed{y\,\,\,\, = \,\,\,\,mx\,\,\, + \,\,\,\,c}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,:{\;\mathbf{Slope}}{\text{ }} - {\text{ }}{\mathbf{intercept}}{\text{ }}{\mathbf{form}}  \\&\qquad\qquad\quad\quad\nearrow \qquad\nearrow \,\,\,\,\,  \\&\qquad\qquad{\text{Slope}}\,\,\,\,\,\,\,\,\,\,\,y - {\text{intercept}}\,\,\,\end{align} \]

This is the general equation of a straight line involving its slope and its y-intercept. This form of the equation of the line is therefore termed the slope-intercept form.

Notice that if the line passes through the origin, its equation would reduce to y = mx.

As you might have guessed by now, this is not the only form to represent a straight line. This form uses the slope and the intercept of the line.

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