I have been looking everywhere for the name of this form of equation of a line $y=m(x-x_0)+y_0$. It's not quite point-slope. It's the same form you would write in the linear approximation of a function, $L(x) = f'(x_0)(x-x_0)+f(x_0)$. It seems important enough to warrant its own name. Does anyone have a clue ?

  • 2
    $\begingroup$ As all of the answers agree, there may not be another name for this form. $\endgroup$ – Sue VanHattum Oct 6 at 17:50

Wolfram calls it the "point slope form", you just have $y_0$ on the rhs rather than lhs.


Several Pearson calculus textbooks, such as Calculus: Graphical, Numerical, Algebraic for high school and Calculus and Analytic Geometry for university use what you show for the point-slope form. The latter puts the $y_0$ term first, but I believe the former puts it last as you do.


Definition of slope of a line, given two known points on the line, say $(x_2,y_2), \;(x_1, y_1)$, where $m$ is the slope of the line:

  • $\dfrac {y_2 - y_1}{x_2-x_1} = m$.

    Note that this is equivalent to the equation: $(y_2-y_1) = m(x_2-x_1)$, which bears close resemblance to the next form of an equation.

Once one calculates slope, or it is given, one can use either of the following point-slope forms of the equation of a line, and one of the points on the line, call it $(x_0, y_0)$:

  • $(y-y_0) = m(x-x_0)$

  • $y= m(x-x_0) - y_0$ (This is algebraically equivalent to the former, in that one merely subtracts $y_0$ from each side of the point-slope form of a line.)

If one knows the slope of a line, $m$, and $b$ (the y-intercept of the line given by the $y$-value of the line when $x=0$, i.e., $(0, b)$), one can use the slope-intercept form of an equation, given by:

  • $y= mx+b$

    This is particularly helpful when the slope of the line is determined, and one point on the line is given by $(0, b)$ is known.

Finally, what is often called the standard form of the equation of a line, is given by:

  • $ax + by=c$.

    This equation is equivalent to the point-slope form of the equation given by $$y = m(x-x_0) - y_0 \\ \\ \iff (\frac 1m)y= x - x_0 - (\frac 1m)y_0\\ \\ \iff x+ (-\frac ym) = \frac{y_0}m-x_0\\ \\ \iff a=1, b =-\frac 1m, c= \frac{y_0}m - x_0.$$ More generally, $(a,b,c)$ can be any ordered triple of the form $\Big(d(1), d\left(-\frac 1m\right), d\left(\frac{y_0}{m}-x_0\right)\Big),$ where $d$ is any constant.

One of the reasons I catalogue each of the above forms of equations of a line is to make evident, that given any of the equations given above, including the equation for calculating slope, if needed, a student should be able to obtain any other form of the equation of a line.

  • $\begingroup$ Interestingly whilst ax+by=c is called the 'standard form', my experiance is the most commonly taught and therefore probably what many people would consider the normal or standard form is y=mx+c $\endgroup$ – David Waterworth Oct 7 at 3:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.