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This is a comparison and is therefore a ratio . Ratios can be expressed as fractions. Thus, a measure of the steepness of a line can be expressed as a ratio.
The slope of a line is defined as the ratio
Mathematically, we can write these changes as
For the two given points, find the slope of the line that passes through them.
and
.
Looking left to right on the line we can choose
to be
, and
to be
Then,
This line has slope 2. It appears fairly steep. When the slope is written in fraction form,
, we can see, by recalling the slope formula, that as
changes 1 unit to the right (because of the
)
changes 2 units upward (because of the
).
Notice that as we look left to right, the line rises.
and
.
Looking left to right on the line we can choose
to be
and
to be
Then,
This line has slope
. Thus, as
changes 2 units to the right (because of the
),
changes 1 unit upward (because of the
).
Notice that in examples 1 and 2, both lines have positive slopes,
and
, and both lines
rise as we look left to right.
and
.
Looking left to right on the line we can choose
to be
and
to be
. Then,
This line has slope
When the slope is written in fraction form,
, we can see that as
changes 1 unit to the right (because of the
),
changes 1 unit downward (because of the
).
Notice also that this line has a negative slope and declines as we look left to right.
and
.
This line has 0 slope. This means it has
no rise and, therefore, is a horizontal line. This does not mean that the line has no slope, however.
and
.
This problem shows why the slope formula is valid only for nonvertical lines.
Since division by 0 is undefined, we say that vertical lines have undefined slope. Since there is no real number to represent the slope of this line, we sometimes say that vertical lines have
undefined slope , or
no slope .
Find the slope of the line passing through and . Graph this line on the graph of problem 2 below.
Find the slope of the line passing through
and
. Graph this line.
The line has slope .
Compare the lines of the following problems. Do the lines appear to cross? What is it called when lines do not meet (parallel or intersecting)? Compare their slopes. Make a statement about the condition of these lines and their slopes.
The lines appear to be parallel. Parallel lines have the same slope, and lines that have the same slope are parallel.
Before trying some problems, let’s summarize what we have observed.
The equation is called the slope-intercept form of the equation of a line. The number is the slope of the line and the point is the .
The slope, of a line is defined as the steepness of the line, and it is the number of units that changes when changes 1 unit.
The formula for finding the slope of a line through any two given points
and
is
The fraction represents the
As we look at a graph from left to right, lines with positive slope rise and lines with negative slope decline.
Parallel lines have the same slope.
Horizontal lines have 0 slope.
Vertical lines have undefined slope (or no slope).
For the following problems, determine the slope and of the lines.
For the following problems, find the slope of the line through the pairs of points.
Do lines with a positive slope rise or decline as we look left to right?
Do lines with a negative slope rise or decline as we look left to right?
decline
Make a statement about the slopes of parallel lines.
For the following problems, determine the slope and of the lines. Round to two decimal places.
For the following problems, find the slope of the line through the pairs of points. Round to two decimal places.
( [link] ) Simplify .
( [link] ) Solve the equation .
( [link] ) When four times a number is divided by five, and that result is decreased by eight, the result is zero. What is the original number?
10
( [link] ) Solve if .
(
[link] ) Graph the linear equation
.
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