Let’s call the coordinates of the first point (x1,y1) , and the coordinates of the second point (x2,y2) . From that, let’s find the slope of the line. Recall that slope is defined as “rise over run”. In other words, it is the amount that y increases for a given amount of x increase. Think of it as stairs. How far up do you go for each step forward? So it is just the difference between the y coordinates divided by the difference between the x coordinates (be sure to preserve the order). We will call the slope m (for historical reasons). So we have:
m=y2−y1x2−x1
Great! So now, how do we use that slope to find the general linear equation? Let’s define some arbitrary point on the line as having coordinates (x,y) . With that, we can pull the same definition of slope we used before, but now using one of the previously defined points, and our new arbitrary point:
m=y−y1x−x1
But since both of these expressions equals the slope, they must equal each other:
m=y2−y1x2−x1=y−y1x−x1
Simplifying this, we get:
y−y1=y2−y1x2−x1(x−x1)
⟹y=y2−y1x2−x1x+y1−y2−y1x2−x1x1
Great! That’s the answer to the first question. Now for the second:
Second question: “How would you create a linear equation if you were given an initial value, and a rate of change?”
So, from the words used, we are talking about how position changes with time. So we can substitute s (position) for y , and t (time) for x , and we are back to discussing the same thing we answered in the first question. We are given a “rate of change”, which is just a fancy term for “slope” when time is the independent variable. So this time we are given the slope. Cool! That will make the final expression simpler. Let’s call that rate of change r . In addition, we are given “an initial value”. That is just a fancy way of saying “the value of s when t=0 ". So lets call this value s0 . Then that means we know that a point on the line is (0,s0) . Great! So just like in the second half of the first answer, we now have a point and a slope. So lets define and arbitrary point on the line as (t,s) , and then use the exact same approach to find the equation:
r=s−s0t−0
⟹s−s0=rt
⟹s=rt+s0
And we are done!
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