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.
Example
Equation of a Line
Find the equation of the line through (0, -3) with slope 3/4.
Solution
We can use the definition of slope, letting (x1, y1)
= (0, -3) and (x, y) represent another point on the line.
A generalization of the method of Example 2 can be used to
find the equation of any line, given its y-intercept and slope.
Assume that a line has y-intercept b, so that it goes through the
point (0, b). Let the slope of the line be represented by m. If (x,
y) is any point on the line other than (0, b) then the definition
of slope can be used with the points (0, b) and (x, y) to get
This result is called the slope-intercept form of the equation
of a line, because b is the y-intercept of the graph of the line.
Slope-intercept form
If a line has slope m and y -intercept b , then the equation
of the line in slope-intercept form is
y = mx + b
When b = 0 we say that y is proportional to x
.
Example
Slope-Intercept Form
Find the equation of the line in slope-intercept form having y
-intercept 7/2 and slope -5/2
Solution
Use the slope-intercept form with b = 7/2 and m = -7/2.
The slope-intercept form shows that we can find the slope of a
line by solving its equation for y. In that form the coefficient
of x is the slope and the constant term is the y-intercept. For
instance, in Example 2 the slope of the line 3x = 4y + 12 was
given as 3/4. This slope also could be found by solving the
equation for y.
4y + 12 = 3x
4y = 3x - 12
The coefficient of x, 3/4, is the slope of the line. The
y-intercept is -3.
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