Equations of a Line - Point-Slope Form
Equations of a Line
An equation in two first-degree variables, such as has a line
as its graph, so it is called a linear equation. In the rest of
this section, we consider various forms of the equation of a
line. 4 x + 7 y = 20, has a line as its graph, so it is called a linear
equation. In the rest of this section, we consider
various forms of the equation of a line.
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The slope-intercept form of the equation of a line involves
the slope and the y-intercept. Sometimes, however, the slope of a
line is known, together with one point (perhaps not the
y-intercept) that the line goes through. The point-slope form of
the equation of a line is used to find the equation in this case.
Let (x1, y1) be any fixed point on the line
and let (x, y) represent any other point on the line. If m is the
slope of the line, then by the definition of slope,
or
y - y1 = m(x - x1)
Point-slope form
If a line has slope m and passes through the point (x1,
y1), then an equation of the line is given by
y - y1 = m(x - x1)
the point-slope form of the equation of a
line.
Example
Point-Slope Form
Find an equation of the line that passes through the point (3,
-7) and has slope m = 5/4.
Solution
Use the point-slope form.
The equation of the same line can be given in many forms. To
avoid confusion, the linear equations used in the rest of this
section will be written in slopeintercept form, y = mx + b, which
is often the most useful form.
The point-slope form also can be useful to find an equation of
a line if we know two different points that the line goes
through. The procedure for doing this is shown in the next
example.
Example
Using Point-Slope Form to Find Equation
Find an equation of the line through (5, 4) and (-10, -2).
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Solution
Begin by using the definition of slope to find the slope of
the line that passes through the given points.
Either (5, 4) or (-10, -2) can be used in the point-slope form
with m = 2/5. If (x1, y1) = (5, 4) then
y - y1 = m(x - x1)
Check that the same result is found if (x1, y1)
= (-10, -2).
Example
Horizontal Line
Find an equation of the line through (8, -4) and (-2, -4).
Solution
Find the slope.
Choose, say, (8, -4) as (x1, y1).
| y - y1 = m(x - x1) |
|
| y - (-4) = 0(x - 8) |
Let y1 = -4, m = 0, x1
= 8 |
| y + 4 = 0 |
0(x - 8) = 0 |
| y = -4 |
|
Plotting the given ordered pairs and drawing a line through
the points, show that the equation y = -4 represents a horizontal
line. See Figure 5(a). Every horizontal line has a slope of zero
and an equation of the form y = k where k is the y-value of all
ordered pairs on the line.
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Example
Vertical Line
Find an equation of the line through (4, 3) and (4, -6).
Solution
The slope of the line is
which is undefined. Since both ordered pairs have x-coordinate
4, the equation is x = 4. Because the slope is undefined, the
equation of this line cannot be written in the slope-intercept
form.
Again, plotting the given ordered pairs and drawing a line
through them show that the graph of x = 4 is a vertical line. See
Figure 5(b).
The slope of a horizontal line is 0.
The slope of a vertical line is undefined.
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