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Math 101 Review Sheet for Exam #2
3 All about lines
3.1 The Rectangular Coordinate System
Know how to plot points in the rectangular coordinate
system.
Know the definitions of x and y intercepts for a line . In particular,
know that these are actually points, and so you need to have two
values . For example, the xintercept of the line described by
5x − 2y = 10
is the point (2, 0); the yintercept for the same line is
the point
(0,−5).
Given an equation for a line , know how to plot it. In
addition
to ‘normal’ lines, you should be able to handle plotting vertical and
horizontal lines.
Try problems 35, 39, 41, 45, 47 and 49, or any additional
problems
from out of 3350 if you feel the need.
3.2 The Slope of a Line
Know the definition of slope, and what geometric
significance it
carries with it. Given two points on a line, or given an equation of
a line, you should be able to compute its slope. You should know
what a slope of 0 and undefined slope means.
For example, any of 2937 should be trivial to find the
slope of
the line.
Given a point on a line, and the slope of a line, you
should be
able to sketch a graph of the line. The point tells you where to
start, and the slope tells you how to find another point on the line.
(Think rise/run).
If you need it, try 38, 41, 42, 43, 44, 45 for practice.
Given two lines, you should be able to determine if they
are
parallel, perpendicular or neither. Questions to think about are 1.)
How do you find the slope of a parallel line? 2.) Given a line, how
do you find the slope of a perpendicular line?
Problems that ask you about these ideas are 4960 any.
3.3 Linear Equations in Two Variables
This looks like the heart of the chapter. Given two pieces
of information
about a line, you should be able to write down an equation
for that line. For example, given two points on the line, or one point
on the line and the slope of the line, you should be able to write
down an equation for the line.
Oftentimes the most useful way to get an equation out of a
couple
pieces of information from a line is to use PointSlope form:
y − y_{1} = m(x − x_{1})
where (x_{1}, y_{1}) is any point on the line, and m is the
slope of the line.
Try problems 19, 2338 any, 41, 43, 4960 any, 61, 65.
3.4 Linear Inequalities in Two Variables
Know the three step process for graphing an inequality. 1)
graph
the boundary line. (dashed if <, solid if the inequality is ≤. 2)
Choose a test point. 3) Shade the appropriate region. When you
write these problems up, be sure that the reader can follow your
work. For example, it is extremely useful to write the words ‘Use
the Test Point (0,0).’
Look at problems 718 any.
4 What Can We Do with Two Lines?
4.1 Two Lines and their intersections
Given two lines, we’d like to look at what happens when we
try
to ‘solve’ both of them simultaneously . Geometrically what this
means is we’d like to find any and all points the two lines have
in common . Algebraically, this means we’d like to find all ordered
pair of numbers , (x, y) such that (x, y) is a solution to the algebraic
equation.
Look at the chart on page 226, because it helps to
understand the
geometric significance of what happens with you run into what are
called the ‘degenerate’ cases. These are the two oddball cases where
the lines happen to be parallel. If this is the case, they are either
the same line, or they never intersect. Algebraically sometimes this
isn’t obvious!
We have two ways to solve for a system of linear
equations. The
first is called elimination, and the second is called substitution . My
guess is that the exam problem testing on this won’t tell you which
method to use , so it’s up to you to pick which method is easier for
the situation.
Good problems to look at are 1732 any and 3954 any. Make
sure you do enough of these until you run into at least on or both
of the degenerate cases (parallel lines).
4.2 Planes
We skipped this section
4.3 ‘Applications’ of Systems of Linear Equations
The exam problem won’t be as nice as the book is. I mean
that
the exam won’t have a table set up for you to fill in. So, in order
to figure out what table to use, we try and mimic some formula we
know , such as d = rt, or amt acid = concentration x volume.
Look at problems 19, 27, 31 and 35.
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