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 x-intercept of the line described by

5x − 2y = 10

is the point (2, 0); the y-intercept 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 33-50 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 29-37 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 49-60 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 Point-Slope form:

y − y₁ = m(x − x₁)

where (x₁, y₁) is any point on the line, and m is the slope of the line.
Try problems 19, 23-38 any, 41, 43, 49-60 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 7-18 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 17-32 any and 39-54 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.