Hello. I'm Professor Von Schmohawk and welcome to Why U.
So far we have studied three common forms of linear equations.
Any of these forms can be used to describe any non-vertical line in the xy-plane.
The most convenient form to use depends upon what information is known about the line.
The slope-intercept form contains the constants "m" and "b"
which specify the line's slope and y-intercept.
The point-slope form also uses "m" to specify slope
but instead of the line's elevation being determined by the y-intercept
the elevation is determined by constants x-one and y-one
the coordinates of any point which lies on the line.
The two-point form, unlike the other two forms
does not contain a constant which specifies the line's slope.
However, like the point-slope form, it includes the coordinates of one point on the line
constants x-one and y-one
plus the coordinates of a second point x-two and y-two
which also lies on the line.
In this lecture, we will introduce one additional and important form of linear equation
called the "standard form".
The standard form has several advantages over the other linear forms we have studied.
One advantage is that, unlike the other forms, the standard form can describe vertical lines.
Therefore, any line in the xy-plane
can be described by a linear equation written in standard form.
The standard form for the equation of a line in the xy-plane
is usually written as "Ax + By = C".
Individually, the constants "A", "B", and "C" have no special meaning.
However, in combination
these constants can tell us a number of things about a line's graph.
For example, the constants "A" and "B" together determine the line's slope.
We can show that this is true by rearranging the standard form
to look like the point-slope form
and seeing which part of the equation corresponds to the slope "m".
We start by subtracting the term "Ax" from both sides
which eliminates the "Ax" term on the left.
We then divide both sides by "B"
and cancel the B's in the numerator and denominator of the fraction on the left.
We then separate the fraction on the right into two fractions with the same denominator
and swap the positions of the two fractions
so that the x term is on the left and the forms look more similar.
Finally, we move the x in the numerator outside the fraction.
Comparing these two forms
we can see that the top equation is now written in point-slope form
except instead of the constants "m" and "b" the top equation contains constant expressions
formed from combinations of the constants "A", "B", and "C".
In particular, the slope "m"
corresponds to the constant expression "negative A over B".
Therefore, for linear equations written in standard form
"negative A over B" is the slope of the line.
We also see that "b" in the slope-intercept form
which determines the y-intercept
corresponds to the constant expression "C over B" in the standard form.
So "C over B" is the value of the y-coordinate of the y-intercept.
We now have expressions which tell us the slope and y-intercept of a line
written in standard form.
But there is another method by which we can determine both the x and y-intercepts.
We know that the y-intercept's x-coordinate is always zero.
This is because the y-intercept is the point where the graph intersects the y-axis
and every point on the y-axis has an x-coordinate of zero.
So another way of determining the y-intercept
is to set x to zero in the equation and see what the resulting value of y is.
Setting x to zero causes the x term to vanish.
Then dividing both sides by "B"
we see once again, that the y-coordinate of the y-intercept is equal to "C over B".
We can determine the x-intercept in a similar fashion.
Since every point on the x-axis has a y-coordinate of zero
we can set y to zero causing the y term to vanish
and solve the equation for x.
So the x-coordinate of the x-intercept is "C over A".
We now have expressions for the x and y-intercepts of a line written in standard form.
One advantage to using the standard form is that it is especially easy to graph a line
since we can quickly determine both its x and y-intercepts.
As an example, in the equation 5x - 3y = 15
the constants "A", "B", and "C" are 5, -3, and 15.
So the x-intercept, "C over A, zero"
has an x-coordinate of "C over A"
which is 15 divided by 5
or 3.
And the y-intercept, "zero, C over B"
has a y-coordinate of "C over B"
which is 15 divided by -3
or -5.
Now to graph this equation, we simply draw a line that passes through both intercepts.
By setting either "A" or "B" to zero
linear equations written in standard form can represent horizontal
or vertical lines.
When "A" is zero
the x term vanishes
and we have an equation with a single variable, y.
If we divide both sides by "B"
we can see that y is equal to the constant expression "C over B".
Since x can have any value
while y always has the constant value "C over B"
this equation describes a horizontal line whose y-intercept is the point "zero, C over B".
On the other hand if instead of "A" being zero, "B" is zero
then the y term vanishes
and we have an equation with only the variable, x.
Dividing both sides by "A"
we see that x is equal to the constant expression "C over A".
Since y can have any value
while x always has the constant value "C over A"
this equation describes a vertical line with x-intercept "C over A, zero".
So when "A" or "B" is zero
the graph is either a horizontal line or a vertical line.
We have seen several advantages of writing linear equations in standard form.
However, the most important reason for using the standard form
is that it enables systems of multiple linear equations to be solved
a topic which will be discussed later in these lectures.
In the next lecture we will examine various types of real-world problems
which can be solved using linear equations.