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Linear
Equations
Definitions
A linear equation in
two variables has the form y = mx + b for some constants m and b.
The graph is a line with slope m and y-intercept (0,b). It
is defined for all real numbers x since any real number can be substituted
for x . For any non-horizontal line, the y values also take
on all real numbers.
Any linear
equation has a constant rate of increase or decrease. This constant
rate of change is the slope of the line and is represented by m
in the equation. Use the following interactive example to explore
how the graph of the line y = mx + b changes as the values
of the coefficients m and b change. Note any useful observations in
your journal.
Slope of a Line
The slope is the constant
rate of change of a linear equation. It can be thought of as the ratio
of the vertical change to the horizontal change between two points
on the graph of a line. If the two points are (x_{1},
y_{1}) and (x_{2}, y_{2}), then the vertical
change is y_{2} - y_{1} and the horizontal change
is x_{2} - x_{1}. Hence we can use the formula
_{}
to determine the slope
of a line if we know two points on that line. See Figure 1 below.
Explore
the following interactive example to see how the slope of a line is
computed when two points are known. What observations can you make?
In general, when the slope
of a line is positive, the line will be increasing. That is, the graph
will be a line that rises from left to right. When the slope is negative,
the line will be decreasing and the graph will be a line that falls
from left to right. If the slope of a line is 0, then it is neither
increasing nor decreasing, but is constant. The graph of a constant
is a horizontal line.
Use the next
interactive example to see how different values of the slope, m, affect
the graph of the line y = mx + b.
Horizontal and Vertical
Intercepts
The points where a line
crosses the vertical and horizontal axes are known as the vertical
and horizontal intercepts. (These points are often referred to as
the x-intercept and the y-intercept.) Given a linear functiony = mx
+ b,
- The vertical intercept
(y-intercept) is found by evaluating the equation when the input
variable, x, is 0 and is always the same as the constant b.
The horizontal intercept
(x-intercept) is the value of the variable x when the y value
is 0. It is found by solving the equation 0 = mx + b.
Interactive
Example
Now explore
how the values of the y-intercept, b, affect the graph of the line
y = mx + b.
Algebraic
Example
Find the vertical and horizontal
intercept of the line y = -7.2x + 250.
Solution
When x = 0, y
= -7.2(0) + 250 = 250, so the vertical intercept is 250. This
means that the graph crosses the horizontal axis at the point (0,
250). Also notice that this is the value of b in the general
form y = mx + b.
To find the horizontal
intercept we can replace y with 0 and solve the linear equation
0 = -7.2x + 250. The solution is given below.
0 = -7.2x + 250
-250 = -7.2x
34.7 = x
The horizontal intercept
is 34.7. This is the point (34.7, 0) on the graph of the line. A graph
of the line is shown in Figure 2.
Figure 2