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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, Ø.

 

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, Ø.