1. Recall:
A linear equation in two variables x and y has the form
Ax+By=C, where A, B, and C are real numbers and not both A and B are zero.
This is also called a first-degree equation because the
exponents on the variables are all one.
A solution to Ax+By=C is an ordered pair of real numbers
(a, b) such that when a is plugged in for x and b is plugged in for y a true
statement is obtained. There are infinitely many solutions to this equation.
(Plug any number in for one variable and solve for the other .)
Ordered pairs are graphed on a rectangular coordinate
system (or Cartesian coordinate system ). The first number of the ordered pair is
found on the horizontal axis. Then move up or down vertically the amount given
in the second number in the ordered pair.
The graph of Ax+By=C is a line. Every ordered pair on the
line is a solution and every solution is a point on the line.
The equation Ax+By=C is called the standard form for the
equation of a line. Every line can be put in this form.
To graph a line:
Graph by plotting at least two points on the line (but
three is better).
Nice points to graph are the x- and y-intercepts, if these
points exist and are not the same point. The x- intercept is found by letting y =
0. The y-intercept is found by letting x = 0.
The graph of the line with equation Ax+By=0 is a line
passing through (0, 0). To graph this line, find at least one other point, but
two more is better.
A vertical line has equation x = a. (Undefined slope and
x-intercept
(a, 0).)
A horizontal line has equation y = b. (Slope = m = 0 and
y-intercept
(0, b).)
The slope of a line passing through the points (x1,y1)
and (x2,y2) is
A line that goes up from left to right has a positive
slope .
A line that goes down from left to right has a negative slope .
The slopes of parallel lines are equal.
The slopes of perpendicular lines are negative reciprocals. (Their product is
–1.)
2. A line segment is a piece of a line. It may be
represented by giving the coordinates of the endpoints of the line segment.
3. The midpoint formula: (See page 472.) The midpoint of a
line segment with endpoints (x1,y1) and (x2,y2) is the
point
Note the plus sign in each numerator .
The average of the x-coordinates of the endpoints is the
x-coordinates of the midpoint and the average of the y-coordinates of the
endpoints is the y-coordinates of the midpoint.
7.2 Review of Equations of Lines; Linear Models
1. The slope-intercept form for the equation of a line is
y=mx+b.
The slope is m, the coefficient of the x. The y-intercept
is the constant b.
This is the form of the equation that must be used to graph the line in the
calculator .
2. The point-slope form for the equation of a line is y−y1=m(x−x1).
The slope is m and the given point is (x1,y1).
This form of the equation of the line is useful for finding the equation of a
line given a point and the slope or given two points.
3. The vertical line through the point (a, b) has the
equation x=a.
The horizontal line through the point (a, b) has the equation x=b.
4. The standard form for the equation of a line is
where A and B are not both zero and A , B and C are all real numbers. (We will
use the convention that A, B and C are the smallest possible integer
coefficients and
.)
5. We will use linear equations to describe or model real
data if the set of data values changes at a constant rate (or almost at a
constant rate).
6. Linear Models: If we know that the data changes at a
constant (or almost constant) rate, this rate of change is the slope of the
line.
The rate of change is the slope m.
A fixed amount is the y-intercept b. (This b value is also
the value of y if x is zero.)
Then use the form y=mx+b to model the data.
7. If you are given two points on the line, or a point and
the rate of change (the slope) use the form y−y1=m(x−x1)
to model the data.
