Gibilisco S. Everyday Math Demystified: A Self-Teaching Guide

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34. One millionth of a kilometer is equal to

(a) 1 meter

(b) 10 meters

(d) 10 millimeters

(e) 1 micrometer

35. In a graph intended to show the correlation between two variables,

strong negative correlation is indicated by points that

(a) all lie near a horizontal line

(b) all lie near a vertical line

(d) all lie tightly clustered near the center of the graph

(e) are arranged in some manner other than the four described

above

36. Suppose you are performing calculations on a computer, and you

end up with the expression 5.000000000E45. This is

(a) a number between 0 and 1

(b) a number greater than 1

(d) a number less than 1

(e) a number too large to be deﬁned

37. The reciprocal of 5.000000000E45 can be expressed as

(a) 2  10

44

(b) 2  10

(d) 5  10

(e) none of the above

38. In Fig. Test 1-5, the solid curve represents an approximation of stock

price versus time, based on points plotted

(a) continuously

(b) at regular time intervals

Fig. Test 1-4. Illustration for Part One Test Question 32.

Test: Part 1 101

(d) at random

(e) by extrapolation

39. In Fig. Test 1-5, suppose stock price is considered as a function of

time. We can call this

(a) a nondecreasing function

(b) a nonincreasing function

(d) neither a nondecreasing nor a nondecreasing function

(e) a relation, but not a legitimate function

40. In the hexadecimal number system, the single-digit numerals are

(a) 1 and 2

(b) 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10

(d) 1, 2, 3, and 4

(e) none of the above

Fig. Test 1-5. Illustration for Part One Test Questions 38 and 39.

PART 1 Expressing Quantities

102

PART TWO

Finding Unknowns

103

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CHAPTER

Basic Algebra

Linear equations have variables that are not raised to any power (except 1 or 0).

They are the simplest equations to solve, although when there are four or more

linear equations in four or more variables, the solution process can be messy.

In this chapter, we’ll deal with linear equations in one and two variables.

Single-Variable Linear Equations

The objective of solving a single-variable equation is to get it into a form

where the expression on the left-hand side of the equality symbol is exactly

equal to the variable being sought (for example, x), and a deﬁned expression

not containing that variable is on the right.

ELEMENTARY RULES

There are several ways in which an equation in one variable can be manipu-

lated to obtain a solution, assuming a solution exists. Any and all of the

105

principles outlined in Chapter 1 can be applied toward this result. In addi-

tion, the following rules can be applied in any order, and any number of

times.

Addition of a quantity to each side: Any deﬁned constant, variable, or

expression can be added to both sides of an equation, and the result is

equivalent to the original equation.

Subtraction of a quantity from each side: Any deﬁned constant, variable,

or expression can be subtracted from both sides of an equation, and the result

is equivalent to the original equation.

Multiplication of each side by a nonzero quantity: Both sides of an equation

can be multiplied by a nonzero constant, variable, or expression, and the

result is equivalent to the original equation.

Division of each side by a quantity: Both sides of an equation can be divided

by a nonzero constant, by a variable that cannot attain a value of zero, or

by an expression that cannot attain a value of zero over the range of its

variable(s), and the result is equivalent to the original equation.

BASIC EQUATION IN ONE VARIABLE

Consider an equation of the following form:

ax þ b ¼ cx þ d

where a, b, c, and d are real numbers, and a 6¼ c. This equation is solved as

follows:

ax þ b ¼ cx þ d

ax ¼ cx þ d  b

ax  cx ¼ d  b

ða  cÞx ¼ d  b

x ¼ðd  bÞ=ða  cÞ

STANDARD FORM

Any single-variable linear equation can be reduced to this form, called the

standard form:

ax þ b ¼ 0

PART 2 Finding Unknowns

106

where a and b are constants, and x is the variable. The value on the right-

hand side of the equals sign is always 0 when a linear equation is in standard

form.

ILLUSTRATING THE SOLUTION

Suppose we substitute y for 0 in the standard form of a linear equation.

This produces a relation where y is the dependent variable and x is the

independent variable:

ax þ b ¼ y

Let’s create a graph of this relation for particular values of a and b, say a ¼ 2

and b ¼ 3, so the linear equation is

2x þ 3 ¼ y

Put the x-values on the horizontal axis and the y-values on the vertical axis.

Plot several test points by setting x at certain values, calculating the y values

that result, and then plotting the points on the graph. You should get

something that looks like Fig. 5-1. The graph is a straight line. Try this for

several diﬀerent combinations of constants a and b. You will ﬁnd that

the graph always turns out as a straight line. That’s where the term ‘‘linear

equation’’ comes from!

The solution to the single-variable equation ax þ b ¼ 0 is indicated by the

point where the straight-line graph for ax þ b ¼ y crosses the x axis (that is,

Fig. 5-1. When the equation 2x þ 3 ¼ y is plotted as a graph, it turns out to be a straight line.

CHAPTER 5 Basic Algebra 107

the horizontal axis). That is, it is the value of x for the particular case where

y ¼ 0. In Fig. 5-1, this happens to be x ¼3/2.

PROBLEM 5-1

Solve the following equation for x. Show the results on a graph similar

to Fig. 5-1, by replacing the 0 on the right-hand side of the equation by

the variable y.

4x þ 3 ¼ 0

SOLUTION 5-1

To solve 4x þ 3 ¼ 0, proceed like this:

4x þ 3 ¼ 0

4x ¼3 ðsubtract 3 from each sideÞ

x ¼3=4 ðdivide each side by 4Þ

The graph of the equation 4x þ 3 ¼ y is shown in Fig. 5-2. Note where the

straight line crosses the x axis.

PROBLEM 5-2

Solve the following equation for x. Show the results on a graph similar

to Fig. 5-1, by reducing the equation to the standard form ax þ b ¼ 0,

and then replacing the 0 on the right-hand side of the equation by the

variable y.

7x  4 ¼ 3x þ 1

Fig. 5-2. Illustration for Problem 5-1.

PART 2 Finding Unknowns

108

SOLUTION 5-2

To solve 7x  4 ¼ 3x þ 1, proceed like this:

7x  4 ¼ 3x þ 1

7x ¼ 3x þ 5 ðadd 4 to each sideÞ

4x ¼ 5 ðsubtract 3x from each sideÞ

x ¼ 5=4 ðdivide each side by 4Þ

To reduce the equation to the standard form, proceed this way:

7x  4 ¼ 3x þ 1

4x  4 ¼ 1 ðsubtract 3x from each side Þ

4x  5 ¼ 0 ðsubtract 1 from each sideÞ

The graph of the equation 4x  5 ¼ y is shown in Fig. 5-3. Note where the

straight line crosses the x axis; this shows the solution x ¼ 5/4.

PROBLEM 5-3

Solve the following equation for x. Show the results on a graph similar

to Fig. 5-1, by replacing the 0 on the right-hand side of the equation by

the variable y.

2x  4 ¼ 3x

Fig. 5-3. Illustration for Problem 5-2.

CHAPTER 5 Basic Algebra 109

SOLUTION 5-3

To solve 2x  4 ¼ 3x, proceed like this:

2x  4 ¼ 3x

4 ¼ 5 x ðadd 2x to each sideÞ

4=5 ¼ x ðdivide each side by 6Þ

x ¼4=5 ðswitch sidesÞ

To reduce the equation to the standard form, perform this single step:

 2x  4 ¼ 3x

 5x  4 ¼ 0 ðsubtract 3x from each sideÞ

The graph of the equation 5x  4 ¼ y is shown in Fig. 5-4. Note where

the straight line crosses the x axis; this shows the solution x ¼4/5.

Two-by-Two Linear Equations

A pair of linear equations in two variables, called a two-by-two (2  2) set of

linear equations, takes the following general form:

x þ b

y þ c

¼ 0

x þ b

y þ c

¼ 0

where x and y represent the variables, and the a’s, b’s, and c’s represent

constants.

Fig. 5-4. Illustration for Problem 5-3.

PART 2 Finding Unknowns

110