
Skills to Master
·
Know and use the
point-slope equation of a line to write linear equations given either the point
and slope of the line or two points on the line
·
Know and use the
slope-intercept equation of a line to write equations of a line given either
the slope and y-intercept of the line or the slope and any point on the line
·
Graph equations
given in slope-intercept form
·
Determine the
slope and y-intercept for a line given in standard (general) form
·
Point-Slope form of a Linear Equation
In the previous
section, we used the slope-intercept
form of a linear equation. That form
is
, where m is the slope
of the line and b is the
y-intercept. A slightly different form
is called the point-slope form. It is
, where as before m is
the slope of the line and
and
are the coordinates of a
point on the line. The y and the x are treated as variables.
That is, whereas you would substitute actual number values for
and
, you would leave y
and x as variables when using this
equations.
This form is derived
from the slope formula. If we say that x and y are any points on some line with slope m, and
is some particular point on the line, then we
have:
![]()
We can cancel the
numerator by multiplying both sides:
![]()
Cancelling on the right
gives:
,
which is the same as ![]()
This form allows us to
write the equation of a line when the slope is known and when at least one
point is known.
Example
What is the equation of
the line with slope m = 3 and passing
through the point (4, -1)?
What is the equation of
the line with slope
and passing through the point (-3, 2)?
This equation can also
be used to write the equation of a line when given only two points:
Example
Write the equation of
the line passing through the points (1, -6) and (-2, -9)
Write the equation of
the line passing through the points (4, -2) and (-4, 4)
Slope-Intercept form of a Linear Equation
Yesterday you learned a
bit about the slope-intercept form:
. Today you learned that, in this form, the
value of m is the slope of the
line. Notice that when the value of x is zero, then
. That is, the point associated with
is
. But recall from yesterday that this is the
special point called the y-intercept. Thus, b
is the y-intercept of the line.
IMPORTANT: the
slope-intercept form of a linear equation gives the values of the slope (m) and the y-intercept (b) directly.
Example
Write the equation of
the line with slope
and y-intercept ![]()
Write the equation of
the line with slope
and y-intercept ![]()
We can also use this
form to find the equation of a line when given the slope and a point on the
line, like one of the problems above.
The difference is that using this form produces only the value of b.
Example
Find the equation of
the line with slope m = 2 and passing
through (1, 9)
Find the equation of
the line with slope m = -1 and
passing through (4, -4)
Standard Form of the Equation of a Line
In the previous section
you learned that another form of a linear equation is
,
where A, B, and C are
integers. Unlike the slope-intercept
form, it is not possible to directly
read the values of m and b (the slope and the y-intercept) from
the standard form. However, we can
determine from the equation how to find
these values by solving the equation for y:
![]()
Since we want to
isolate the y, we must first replace
the value of Ax with zero. We do this by subtracting Ax from both sides of the equation. This gives:
![]()
Now we want to replace
the value of B with a 1. We do this by dividing. Remember to divide all terms on both sides by
B!
This gives:
![]()
Compare this to
. Which part is like m? Which part is like b?
We see, then, that the
slope is
and the y-intercept is
.
Example
What is the slope of
the line and the y-intercept for the equation
?
What is the slope of
the line and the y-intercept for the equation
?
To complete this
section, let’s look at two types of problems that require you to use several of
the concepts about linear equations that you’ve studied so far:
Example
What is the equation of
the line that passes through the point (4, 3) and that is parallel to the line
?
What is the equation of
the line passing through the point (-1, -1) and perpendicular to the line
?
Try these:
What is the equation of
the line passing through the point (0, -4) and parallel to the line
?
What is the equation of
the line passing through the point (2, 4) and perpendicular to the line
?