1
C H A P T E R 1
Review of Algebra
is book is a text on calculus, structured to prepare students for applying calculus to the physical
sciences. e first chapter has no calculus in it at all; it is here because many students manage to
get to the university or college level without adequate skill in algebra, trigonometry, or geometry.
We assume familiarity with the concept of variables like x and y that denote numbers whose
value is not known.
1.1 SOLVING EQUATIONS
An equation is an expression with an equals sign in it. For example:
x D 3
is a very simple equation. It tells us that the value of the variable x is the number 3. ere are a
number of rules we can use to manipulate equations. What these rules do is change an equation
into another equation that has the same meaning but a different form. e things we can do to
an equation without changing its meaning include the following.
• Add or subtract the same term from both sides. If that term is one that is already present
in the equation, we may call this
moving the term to the other side
. When this hap-
pens, the term changes sign, from positive to negative or negative to positive. For example:
x 4 D 5 is is the original equation
x D 5 C 4 Add 4 to both sides (move 4 to the other side)
x D 9 Finish the arithmetic
• Multiply or divide both sides by the same expression.
2 1. REVIEW OF ALGEBRA
For example:
3x 4 D 5 is is the original equation
3x D 5 C 4 Add 4 to both sides (move 4 to the other side)
3x D 9 Finish the addition
3x
3
D
9
3
Divide both sides by 3
x D 3 Finish the arithmetic
• Apply the same function or operation to both sides.
For example:
p
x 2 D 5 is is the original equation
p
x 2
2
D 5
2
Square both sides
x 2 D 25 Do the arithmetic
x D 27 Move 2 to the other side
Knowledge Box 1.1
e rules for solving equations include:
1. Adding or subtracting the same thing from both sides.
2. Moving a term to the other side; its sign changes.
3. Multiplying or dividing both sides by the same thing.
4. Performing the same operation to both sides, e.g., squaring or tak-
ing the square root.
ese rules are illustrated by the examples in this section.
1.1. SOLVING EQUATIONS 3
Solving an equation can get very hard at times and much of what we will do in this chapter is
to give you tools for solving equations efficiently.
Example 1.1 Solve the equation 3x 7 D x C 3 for x.
Solution:
3x 7 D x C 3 is is the original equation
2x D 10 Move 7 and x to the other side
x D 5 Divide both sides by 2
˙
All the examples so far have been solving for the variable x. We can solve for any symbol.
Example 1.2 Solve P V D nRT for P .
Solution:
P V D nRT is is the original equation
P D
nRT
V
Divide both sides by V
˙
When we solve an equation for a variable, we put it in functional form. We will learn more
about functions in Section 1.4. When an equation is in functional form, the variable we solved
for represents a value we are trying to find. In Example 1.2, which is the Ideal Gas Law,
this would be the pressure of a gas. We call this the dependent variable. e variable (or
variables) on the other side of the equation represent values that are changing. We call these
the independent variables. In Example 1.2, these are n and T , the amount and temperature of
the gas. (R is the universal gas constant—not a variable.)
4 1. REVIEW OF ALGEBRA
Let’s try a slightly harder example.
Example 1.3 Solve y D
2x 1
x C 2
for x.
Solution:
y D
2x 1
x C 2
is is the original equation
y.x C 2/ D 2x 1 Multiply both sides by .x C 2/
xy C 2y D 2x 1 Distribute y
xy C 2y C 1 D 2x Move 1 to the other side
2y C 1 D 2x xy Move xy to the other side
2y C 1 D x.2 y/ Factor out x on the right
2y C 1
2 y
D x Divide both sides by .2 y/
x D
2y C 1
2 y
Neaten up
˙
Example 1.3 used a strategy. First, clear the denominator; second, get everything with x in it
on one side and everything else on the other side; third, factor to get a single x; and fourth,
divide by whatever is multiplied by x in order to isolate x, obtaining the solution.
Example 1.4 Solve x
2
C y
2
D 25 for y.
Solution:
x
2
C
y
2
D
25
is is the original equation
y
2
D 25 x
2
Move x
2
to the other side
y D ˙
p
25 x
2
Take the square root of both sides
˙
1.1. SOLVING EQUATIONS 5
e original equation in Example 1.4 graphs as a circle of radius 5 centered at the origin.
When we take the square root of both sides, the fact that squaring a number makes it positive
means that the plus and minus square roots are both correct. By convention, if we need a single
expression, we take the positive square root.
PROBLEMS
Problem 1.5 Solve the following equations for x.
1. 7x C 4 D 17
2. 8x 6 D 66
3. 7x C 5 D 68
4. 5x C 3 D 43
5. 5x C 4 D 19
6. 2x 5 D 9
Problem 1.6 Solve the following equations for x.
1.
p
6x C 3 D 15
2.
p
7x C 8 D 13
3.
p
32x C 4 D 6
4.
p
x 1 D 10
5.
p
39x C 3 D 9
6.
p
2x C 3 D 19
Problem 1.7 Solve the following equations for y.
1. x
2
C 14y
2
D 4
2. 2x
2
C 10y
2
D 3
3. 2x
2
C 10y
2
D 20
4. 8x
2
C 17y
2
D 16
5. 8x
2
C 18y
2
D 9
6. 15x
2
C 12y
2
D 8
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