Say we have these two points on a xy plane. (2, 3) and (4, 5). The slope would be 1. To get the equation of the line we can use one of these two formulas.
Substitution y and x values in the formula y = mx + c
3 = 2(1) + C
C = 1
y = 1(x) + 1
Using a different formula y - y1 = m(x - x1)
y - 3 = 1(x - 2)
y = 1(x) + 1
Both of them produce the same results. Is there a scenario where one should be use the other?
On
You even may know by heart the interpolation formula: $$y=\frac{y_2-y_1}{x_2-x_1}(x-x_1)+y_1,$$ which is valid if $x_1\neq x_2$ (trivial case).
On
yes, there are situations where these can be different, for example in programming multiplying by 0.0000001 is faster than dividing by 10000000.
number of operations and type of operations matter, even if the all the values obtained are the same, one can come up with definitions so that the representations can be put into separate categories.
They are essentially the same.
The second equation is $y - y_1 = m(x-x_1)$. Rearranging the terms, we can get $y = mx + (y_1 -mx_1) = mx + C$ where $C = (y_1 -mx_1)$, which then we obtain the first equation.