Types of Linear Equations
We often think of an equation such as 3x + 4x = 7x as an "addition fact"
because the equation is satisfied by all real numbers. However, some equations
that we think of as facts are not satisfied by all real numbers. For example,
is satisfied by every real number except 0 because
is undefined. The equation x + 1 = x + 1 is satisfied by all real numbers
because both sides are identical. All of these equations are called
identities.
The equation 2x + 1 = 7 is true only on condition that we choose x = 3.
For this reason, it is called a conditional equation.
Some equations are false no matter what value is used to replace the
variable. For example, no number satisfies x = x + 1. The solution set to this
inconsistent equation is the empty set, Ã˜.
Identity, Conditional Equation, Inconsistent Equation
An identity is an equation that is satisfied by every number for which
both sides are defined.
A conditional equation is an equation that is satisfied by at least
one number but is not an identity.
An inconsistent equation is an equation whose solution set is the
empty set.
It is easy to classify 2x = 2x as an identity and x = x + 2 as an
inconsistent equation, but some equations must be simplified before they can be
classified.
Example 1
An inconsisten equation and an identity
Solve each equation.
a) 8  3(x  5) + 7 = 3  (x  5)  2(x  11)
b) 5  3(x  6) = 4(x  9)  7x
Solution
a) First simplify each side of the equation:
8  3(x  5) + 7 
= 3  (x  5)  2(x  11) 

8  3x  15 + 7 
= 3  x + 5  2x + 22 
Distributive property 
30  3x 
= 30  3x 
Combine like terms 
This las equation is satisfied by any value of x because the two sides are
identical. Because the last equation is equivalent to the original equation,
the original equation is satisfied by any value of x and is an identity. The
solution set is R, the set of all real numbers.
b) Fist simplify each side of the equation.
5  3(x  6) 
= 4(x  9)  7x 

5  3x + 18 
= 4x  36  7x 
Distributive property 
23  3x 
= 36  3x 
Combine like terms 
23  3x + 3x 
= 36  3x + 3x 
Add 3x to each side 
23 
= 36 
Combine like terms 
The equation 23 = 36 is false for any choice of x. Because these equations
are equivalent, the original equation is also false for any choice of x. The
solution set to this inconsistent equation is the empty set,
Ã˜.