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The
equation y = 2x – 1
that we used as an example
for graphing functions produced
a graph that was a straight
line. This was no accident.
This equation is one example
of a general class of equations
that we call linear equations in
two variables. The two
variables are usually (but
of course don’t have to be)
x and y. The
equations are called linear
because their graphs are straight
lines. Linear equations are
easy to recognize because
they obey the following rules:
- The
variables (usually x
and y) appear only
to the first power
- The
variables may be multiplied
only by real number constants
- Any
real number term may be
added (or subtracted, of
course)
- Nothing
else is permitted!
- This
means that any equation
containing things like x2,
y2, 1/x,
xy, square roots,
or any other function of
x or y is
not linear.
Just as there are an infinite
number of equations that satisfy
the above conditions, there
are also an infinite number
of straight lines that we
can draw on a graph. To describe
a particular line we need
to specify two distinct pieces
of information concerning
that line. A specific straight
line can be determined by
specifying two distinct points
that the line passes through,
or it can be determined by
giving one point that it passes
through and somehow describing
how “tilted” the line is.
The slope of a line
is a measure of how “tilted”
the line is. A highway sign
might say something like “6%
grade ahead.” What does this
mean, other than that you
hope your brakes work? What
it means is that the ratio
of your drop in altitude to
your horizontal distance is
6%, or 6/100. In other words,
if you move 100 feet forward,
you will drop 6 feet; if you
move 200 feet forward, you
will drop 12 feet, and so
on.
We measure the slope of lines
in much the same way, although
we do not convert the result
to a percent.
Suppose we have a graph of
an unknown straight line.
Pick any two different points
on the line and label them
point 1 and point 2:
In moving from point 1 to
point 2, we cover 4 steps
horizontally (the x
direction) and 2 steps vertically
(the y direction):
Therefore, the ratio of the
change in altitude to the
change in horizontal distance
is 2 to 4. Expressing it as
a fraction and reducing, we
say that the slope of this
line is
To formalize this procedure
a bit, we need to think about
the two points in terms of
their x and y
coordinates.
Now you should be able to
see that the horizontal displacement
is the difference between
the x coordinates of
the two points, or
4 = 5 – 1,
and the vertical displacement
is the difference between
the y coordinates,
or
2 = 4 – 2.
In general, if we say that
the coordinates of point 1
are (x1, y1)
and the coordinates of point
2 are (x2, y2),
then we can define the slope
m as follows:
where
(x1, y1)
and (x2,
y2) are
any two distinct points on
the line.
- It
is customary (in the US)
to use the letter m
to represent slope. No one
knows why.
- It
makes no difference
which two points are used
for point 1 and point 2.
If they were switched, both
the numerator and the denominator
of the fraction would be
changed to the opposite
sign, giving exactly the
same result.
- Many
people find it useful to
remember this formula as
“slope is rise over run.”
- Another
common notation is
 ,
where the Greek letter delta
(D) means “the change in.” The slope is a ratio
of how much y changes
per change in x:
A
horizontal line has zero slope,
because there is no change
in y as x increases.
Thus, any two points will
have the same y coordinates,
and since y1 = y2,
 .
A
vertical line presents a different
problem. If you look at the
formula
 ,
you
see that there is a problem
with the denominator. It is
not possible to get two different
values for xl
and x2,
because if x changes
then you are not on the vertical
line anymore. Any two points
on a vertical line will have
the same x coordinates,
and so x2 – x1 = 0.
Since the denominator of a
fraction cannot be zero, we
have to say that a vertical
line has undefined slope.
Do not confuse this with the
case of the horizontal line,
which has a well-defined slope
that just happens to equal
zero.
The
x coordinate increases
to the right, so moving from
left to right is motion in
the positive x direction.
Suppose that you are going
uphill as you move in the
positive x direction.
Then both your x and
y coordinates are increasing,
so the ratio of rise over
run will be positive—you will
have a positive increase in
y for a positive increase
in x. On the other
hand, if you are going downhill
as you move from left to right,
then the ratio of rise over
run will be negative because
you lose height for
a given positive increase
in x. The thing to
remember is:
As
you go from left to right,
- Downhill = Negative
Slope
And of course, no change in
height means that the line
has zero slope.
Some
Slopes
Two lines can have the same
slope and be in different
places on the graph. This
means that in addition to
describing the slope of a
line we need some way to specify
exactly where the line is
on the graph. This can be
accomplished by specifying
one particular point that
the line passes through. Although
any point will do, it is conventional
to specify the point where
the line crosses the y-axis.
This point is called the y-intercept,
and is usually denoted by
the letter b. Note
that every line except vertical
lines will cross the y-axis
at some point, and we have
to handle vertical lines as
a special case anyway because
we cannot define a slope for
them.
Same
Slopes, Different y-Intercepts
The equation of a line gives
the mathematical relationship
between the x and y
coordinates of any point on
the line.
Let’s return to the example
we used in graphing functions.
The equation
y = 2x – 1
produces the following graph:
This line evidently has a
slope of 2 and a y
intercept equal to –1. The
numbers 2 and –1 also appear
in the equation—the coefficient
of x is 2, and the
additive constant is –1. This
is not a coincidence, but
is due to the standard form
in which the equation was
written.
Standard Form (Slope-Intercept
Form)
If a linear equation in two
unknowns is written in the
form
where
m and b are
any two real numbers, then
the graph will be a straight
line with a slope of m
and a y intercept equal
to b.
Point-Slope Form
As mentioned earlier, a line
is fully described by giving
its slope and one distinct
point that the line passes
through. While this point
is customarily the y
intercept, it does not need
to be. If you want to describe
a line with a given slope
m that passes through
a given point (x1,
y1), the
formula is
To
help remember this formula,
think of solving it for m:
Since
the point (x, y)
is an arbitrary point on the
line and the point (x1, y1)
is another point on the line,
this is nothing more than
the definition of slope for
that line.
Two-Point Form
Another way to completely
specify a line is to give
two different points that
the line passes through. If
you are given that the line
passes through the points
(x1, y1)
and (x2, y2),
the formula is
This
formula is also easy to remember
if you notice that it is just
the same as the point-slope
form with the slope m
replaced by the definition
of slope,
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