This is the way I teach the formula for a straight line.
We begin with the formula for the slope of the line
. . through two points: \(\displaystyle (x_1,y_1)\) and \(\displaystyle (x_2,y_2)\!:\;\boxed{m \;=\;\dfrac{y_2-y_1}{x_2-x_1}}\) .[1]
Now "move 2 onto the 1": .\(\displaystyle m \;=\;\dfrac{y-y_1}{x-x_1}\)
Cross-multiply: .\(\displaystyle \boxed{y - y_1 \;=\;m(x-x_1)}\) .[2]
. . This is called the Point-Slope Formula.
With these two formulas, we can write the equation of any line.
If we are given a point \(\displaystyle (x_1,y_1)\) and the slope \(\displaystyle m\),
. . then use formula [2].
If we are given two points \(\displaystyle (x_1,y_1),\: (x_2,y_2)\)
. . use formula [1] to find \(\displaystyle m\), then use formula [2].
If we are given the slope \(\displaystyle m\) and the \(\displaystyle y\)-intercept \(\displaystyle b\),
. . recall that we have a point \(\displaystyle (0,b)\), then use formula [2].
If we are given the \(\displaystyle x\)-intercept \(\displaystyle a\) and the \(\displaystyle y\)-intercept \(\displaystyle b\),
. . we have two points \(\displaystyle (a,0),\: (0,b).\)
. . Use formula [1] to find \(\displaystyle m\), then use formula [2].
Yet I have seen textbooks (and teachers) require \(\displaystyle four\) formulas:
. . \(\displaystyle \begin{array}{c}\text{Point-Slope Formula:} & y - y_1 \:=\:m(x-x_1) \\ \text{Two-Point Formula:} & y - y_1 \;=\;\dfrac{y_2-y_1}{x_2-x_1}(x-x_1) \\ \text{Slope-Intercept Formula:} & y \:=\:mx + b \\ \text{Two-Intercept Formula:} & \dfrac{x}{a} + \dfrac{y}{b} \:=\:1 \end{array}\)
What a waste of brain cells . . .