What is the name of the form of the line equation $y = m(x-x0)+y0$

Question

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 ?

Answer 1

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

Answer 2

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.

Answer 3

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.