Is there any point to doing something the long way when you can just as well do it in a shorter, much more efficient way? I say, Heck no! We can do things in the “Triple-F” way: Fast, Fun & Friendly, and with deep understanding, to boot.
So in that spirit, today I’ll get us started on quickly and effortlessly converting a linear equation from what’s called standard form (Ax + By = C) to what we know as the good-old slope-intercept form (y = mx + b).
To better grasp standard form, let’s replace its mysterious A, B, and C with actual numbers: 4 for A, 2 for B and 8 for C. That gives us the more typical looking equation of an actual line: 4x + 2y = 8. Do you recall seeing this kind of equation in your algebra text and class? Sure you do. You get this kind of equation in the chapter(s) on the coordinate plane and in other spots, too.
Now usually when books teach us how to convert from this “standard” form to slope-intercept form, they tell us to solve the equation for y. Of course that works, but it takes too darn long.
To understand the quicker way, let’s have a little fun with the standard form of the equation: Ax + By = C
We’re going to start with this standard form and solve that for y. And as you’ll see, we’ll learn some useful things from the result.
To kick things off, we start with Ax + By = C, and we subtract the Ax term from both sides. That leaves us with this equation:
By = – Ax + C
Now take this new equation and divide both sides by B. That gives us this little gem of an equation:
y = (– A/B) x + C/B
I’m going to call this the magic equation both to give us a way to refer to it and to show us what’s so useful about it.
The big insight is that this magic equation is actually, believe-it-or-not, in slope-intercept form; we just need to SEE it that way. Here’s how.
In slope-intercept form (y = mx + b), notice that the y variable is all by its lonesome on the left side. Do we have that in the magic equation? Yes we do. So … CHECK!
In slope-intercept form, there’s a value called m (aka, the slope) that is multiplying the x variable. Do we have something in the magic equation that’s multiplying the x variable? Why yes, and it happens to be
(–A/B). So do we have the slope showing in the magic equation? Yes, the slope is: (–A/B). So … CHECK!
Finally, in slope-intercept form, there’s a constant (i.e., a number term, not a variable term) that appears after the mx term. So do we have a constant after the mx term in the magic equation? Yes, indeed. We have C/B. Note that in any actual linear equation, B and C will be actual numbers, not variables. So the value you get when you divide B by C (the quotient B/C), also must be a real number, just as surely as the real numbers 8 and 2 gives us the real number 4 when we divide 8 by 2.
So, to address the final question, do we have a b-value in the magic equation? Yes, it’s (C/B). C/B is the y-intercept, the real number we call b in slope-intercept form. So once again … CHECK.
So all in all, do we now have the equation in slope-intercept form? Yes, indeed. You just need to realize that
(–A/B) is the slope, and (C/B) is the y-intercept.
In my next post I’ll show you how you use these results to quickly transform the equation from standard form to slope-intercept form. It will be amazing.