Section 2.3 ­ Linear Equations from Two Points
We have already talked about how to find the slope of a line. How do you do it?
The formal equation to find the slope is:
Notice that the 1's and 2's are subscripts, not superscripts.
They do not represent exponents. They are there to distinguish the two points.
It does not matter which point you call point 1 and which you call point 2.
If you know that slope is just a ratio comparing rise to run, you DON'T need the equation.
Example:
Find the slope.
x
y
4
5
6
2
8
­1
Our goal is to be able to write an equation of a line from two points .
What are the two things I need to write the equation of the line in slope­intercept form?
If I have two points, how can I find the slope?
Can you think of a way to get the y­intercept?
Finding the Equation of a Line from a Table (or Two Points)
1. Pick any two points and find the slope.
2. If you were not given the y­intercept, pick any point from the table and plug both it and the slope into y = ax + b. This will allow you to solve for b.
Warning: This is one that people have a lot of difficulty with.
Example:
Find the equation of the line represented by the table:
x
y
4
5
6
2
8
­1
Examples:
Find the equation of a line through (­6, 6) and (3, ­9)
Example: Word Problem Sec. 2.3, p. 62, #41
A software company sells educational site licenses for its software. If LBCC purchases the license, the number of manuals provided is dependent upon the number of computers the license allows the software to be installed on. Based upon the table below, write an equation for the number of manuals provided. Graph the equation. Predict the number of manuals that will be provided for 100 computer stations. Number of computer stations
5
10
20
30
50
# of manuals provided
2
4
8
12
20
Example: Word Problem Sec. 2.3, p. 62, #41
Based upon the table below, write an equation for the number of manuals provided. Graph the equation. Predict the number of manuals that will be provided for 100 computer stations. Number of computer stations
5
10
20
30
50
# of manuals provided
2
4
8
12
20
Manuals provided
40
30
20
10
20
40
60
80
100
Computer stations
Take a look at Example 4 on p. 56 of the book.
Make sure you understand the "years since" idea.
Finding the Equation of a Line from its Graph
1. Pick any two points from the graph. Calculate slope (rise / run). Be careful to use correct signs. This is a.
2. Find the vertical intercept or the point where the line intersects the vertical axis. (If the line doesn't go through the y­intercept, extend it.) The coordinates are (0, b).
3. Substitute the values of a and b into the equation
y = ax + b.
Example:
Find the equation of the line.
Example:
Find the equation of the line.
Can you get back forth between graphs, tables, and equations?
Graph
Table
Equation