Professor Dave again, let’s graph some more lines.
We now understand how to graph lines in the coordinate plane, but we have been doing this
exclusively using the Y equals MX plus B form, otherwise known as the slope-intercept form.
There is another form that we will see for lines, and it’s called the standard form,
which is AX plus BY equals C.
This is just another way of communicating linear relationships, and it can be useful
in certain situations, especially if we want to graph a line very quickly.
To see why it’s so easy to graph this way, let’s try an example.
Say we have four X plus six Y equals twelve.
Instead of plugging in X values to make a table, let’s just rationalize that the Y
intercept, or the point where the line crosses the Y axis, must have an X value of zero.
Likewise, the X intercept, or the point where the line crosses the X axis, must have a Y
value of zero.
For this reason, to find the Y intercept, we can just plug in zero for X.
The X term will go away, six Y equals twelve, so Y equals two.
That means the Y intercept is the point zero two.
Then to find the X intercept, we plug in zero for Y.
The Y term will go away, four X equals twelve, and X equals three.
That means the X intercept is the point three zero.
With two points on the line now apparent, we are ready to draw the line, and that’s
all there is to it.
So we should note that standard form is useful when trying to find the intercepts of a line,
and when trying to quickly graph the line.
However, if we want to talk about the slope of the line, it’s a little trickier, and
slope-intercept form is better for that, since the slope is listed right in the equation.
If we have a line in standard form and we want to know the slope, there are two things
we could do.
We could take the two intercepts, which represent points on the line, and use the coordinates
of those points to calculate the slope.
For this example, the two points are zero two and three zero.
That means the rise is negative two and the run is three, which gives us negative two
thirds for the slope.
The other thing we could do is convert this equation into slope-intercept form.
All we have to do is solve for Y.
So the first thing to do would be to subtract four X from both sides.
Now we have six Y equals negative four X plus twelve.
Then, we divide both sides by six.
That gives us Y equals negative four sixths X, which simplifies to negative two thirds
X, plus twelve over six, which is two.
And converting from standard form to slope-intercept form is just that simple.
We could also go in reverse, from slope-intercept to standard form.
To do this, we should note that in standard form, A, B, and C must be integers, they can’t
be fractions, so the first thing we would do is multiply everything by three.
That gives us three Y equals negative two X plus six.
Then we could add two X to both sides to get two X plus three Y equals six.
Now we can see that this form is actually a little simpler than our original equation.
In fact, each term seems to be half of the original term.
What this tells us is that when writing an equation in standard form, not only do A,
B, and C have to be integers, but they should be in the lowest ratio of these integers that
is possible.
Four, six, and twelve can all be divided by two to get a simpler version of the equation,
so this will be the more correct version of the equation.
We would still get the same values for the intercepts and therefore the same line, but
we always want the simplest form of the equation possible, so we must bear this in mind when
converting between the two linear forms.
Now that we have both of these forms down, let’s check comprehension.