Section 1.2 Solving Equations
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Version: Fall 2007
1.2 Solving Equations
In this section, we review the equation-solving skills that are prerequisite for successful
completion of the material in this text. Before we list the tools used in the equation-
solving process, let’s make sure that we understand what is meant by the phrase “solve
for
x
.”
Solve for
x
.
Using the properties that we provide, you must “isolate
x
,” so that
your ﬁnal solution takes the form
x
= “Stuﬀ,”
where “Stuﬀ” can be an expression containing numbers, constants, other variables,
and mathematical operators such as addition, subtraction, multiplication, division,
square root, and the like.
“Stuﬀ” can even contain other mathematical functions, such as exponentials, loga-
rithms, or trigonometric functions. However, it is essential that you understand that
there is one thing “Stuﬀ” must not contain, and that is the variable you are solving
for, in this case,
x
. So, in a sense, you want to isolate
x
on one side of the equation,
and put all the other “Stuﬀ” on the other side of the equation.
Now, let’s provide the tools to help you with this task.
Property 1.
Let
a
and
b
be any numbers such that
a
=
b
. Then, if
c
is any
number,
a
+
c
=
b
+
c,
and,
a
−
c
=
b
−
c.
In words, the ﬁrst of these tools allows us to add the same quantity to both sides
of an equation without aﬀecting equality. The second statement tells us that we can
subtract the same quantity from both sides of an equation and still have equality.
Let’s look at an example.
⚏
Example 2.
Solve the equation
x
+ 5 = 7
for
x
.
The goal is to “isolate
x
on one side of the equation. To that end, let’s subtract 5
from both sides of the equation, then simplify.
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