Section 1-1: Points and Lines
This section is a basic
review of lines and points and their relationship to graphing. It
is your job to know each of these terms, so get to it and memorize them!!
Demo: Points in the xy plane! (Exploremath - requires Shockwave)
Demo: Distance Formula! (Exploremath - requires Shockwave)
Demo: 2x2 Linear systems! (Exploremath - requires Shockwave)
1)
A solution of an equation is an ordered pair
that makes the equation true.
2)
The x - intercept is where the graph crosses
the x-axis (also called zero,
root or solution).
The y - intercept is
of course, where the graph crosses the y-axis.
3)
A linear equation is in the form of Ax + By
= C where A and B are not both zero! This is called the general
form of a linear equation and graphs into
a line. (amazing!)
Sketching the graph of
a line!
One technique for graphing
a line is to find the intercepts. Recall that it takes only two points
to determine a line! To find the y-intercept let x
= 0 and to find the x-intercept let y
= 0!
Sketch the graph of 2x +
5y = 10.
Solution:
Find the y-intercept by
letting x = 0
2(0) + 5y = 10
5y = 10
y = 2 Therefore, the
y-intercept is (0, 2)
Find the x-intercept by
letting y = 0
2x + 5(0) = 10
2x = 10
x = 5 Therefore, the x-intercept
is (5, 0)
Here is the graph!!
Remember, there are three
types of lines: vertical, horizontal and slanted! You need to know
the equations of the horizontal and vertical.
Study the following graphs:
Note that the equation of
a vertical line is x = constant(yellow on graph)
and the equation of a horizontal
line is y = constant(purple on graph)
Intersection of lines
There are three ways two
lines can intersect in a plane. They can intersect exactly at one
point, or they can be parallel, or they can have an infinite number
of solutions (same line).
To find the intersection,
solve the equations simultaneously. (at the same time!)
Example: Find the
intersection point for the equations
2x + 5y = 10
x + y = 5
Using linear combinations
(add/subtract method), multiply #2 by -2
-2x - 2y = -10
2x + 5y = 10
3y = 0
y = 0
To find x, replace y with
0 in either of the original equations.
x + 0 = 5
x = 5
The intersection point is
(5, 0)
The graph is below:
Important to remember 
When using this method,
if both variables are eliminated, the result depends on the truth/falseness
of the statement. If the resulting statement is true, this means
the lines are the same and you have an infinite number of solutions.
But, if the resulting statement is false, the lines have no common solution,
they are parallel!!
Distance and Midpoint Formulas
Let A = (x1,
y1), B = (x2,
y2) and M be the midpoint
of AB. Then:
Example) Find the distance
between (-1, 9) and (4, -3). Then find the midpoint.
Distance = 
Midpoint = 
That does it for the
first section. It is basic algebra review. You should have
no problem with it!! On to the next section
.
Darkwing says, "Press
the button, dude!"