## Kiss those Math Headaches GOODBYE!

### How to quickly find the y-intercept (b-value) of a line

Of course there’s a standard way to find the y-intercept of any line, and there’s nothing wrong with using that approach.

But the method I’ll present here is a bit faster and therefore easer. And hey, if we can save time when doing math, it’s worth it … right?

So first let’s recall that the y-intercept of any function is the y-value of the function when the x-value = 0. That’s because the y-intercept is the y-value where the function crosses or touches the old, vertical y-axis, and of course all along the y-axis the x-value is always 0 (zero).

So the standard slope-intercept formula is y = mx + b. In a problem asking for the y-intercept, you’ll be given one point that the line passes through (that point’s coordinates will provide you with an x-value and a y-value), and you will also be told the slope of the line (the line’s m-value).
So then, to get the b-value, which is the value of the y-intercept, you just grab your y = mx + b equation (dust it off if you haven’t used it in a while), and plug in the three value you’ve been given: those for x, y and m. Then you solve the equation for the one variable that’s left: b, the value of the y-intercept.

Let’s look at an example: a line with a slope of 2 passes through the point (3, 10). What is this line’s y-intercept.

Now, according to the problem, the m-value = 2, the x-value = 3, and the y=value = 10. We just take these values and plug them into the equation:
y = mx + b, like this:

10 = (2)(3) + b

After doing these plug-ins, you just solve the equation for b, finding that
b = 4. That means that the y-intercept of the line = 4.

Now let’s see how you can do the same problem, but a little bit faster.
To do so, we first need to play around with the y = mx + b equation by subtracting the mx-term from both sides, like this:

y = mx + b [Standard equation.]
– mx = – mx [Subtracting mx from both sides.]
y – mx = b [Result after subtracting.]
b = y – mx [Result after flipping left & right sides
of the equation above.]

Aha! Look at that final, beautiful equation. This equation has b isolated on the left-hand side. So now if we want to solve for b, all we do is plug in the x, y and m values into the right-hand side of the equation and simplify the value, and the value we get will be the b-value.

For the problem we just solved, with x = 3, y = 10, m = 2, watch how easy it is to solve:

b = y – mx
b = 10 – (2)(3)
b = 10 – 6
b = 4

So notice that this technique, just like the first technique, reveals that the
y-intercept of the line is 4, or (0, 4). The techniques agree, they just get to the same end in slightly different ways.

Notice that with the second, quicker technique, you don’t need to add or subtract any terms. And that’s a key reason that this technique is faster and easier to use than the standard method. So try it out and stick with it if you like it.

### How to Transform from Standard Form to Slope-Intercept Form

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.

High-Octane Boost for Math Ed

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.