Recognizing Conditional Equations, Identities, and
ContradictionsSolve each equation. Decide whether it is a conditional equation,an identity,or a
contradiction.
(a)
Distributive property
Combine like terms.
Subtract 7x. Subtract 28.
Combine like terms.Divide by 3.The solution set, , has only one element, so is a
conditional equation.
(b)
Distributive property
Subtract 5x. Add 15.
TrueThe final line, , indicates that the solution set is all real numbers , and the equa-
tion is an identity. (The first step yielded
which is truefor all values of x. We could have identified the equation as an identity at
that point.)
(c)
Distributive property
Subtract 5x.
FalseSince the result, is false,the equation has no solution. The solution set
is 0 ,so the equation 5 x- 15 = 51 x- 42 is a contradiction.
- 15 =-20,
- 15 =- 20
5 x- 15 - 5 x= 5 x- 20 - 5 x
5 x- 15 = 5 x- 20
5 x- 15 = 51 x- 42
5 x- 15 = 51 x- 32 5 x- 15 = 5 x-15,
0 = 0 5 6
0 = 0
5 x- 15 - 5 x+ 15 = 5 x- 15 - 5 x+ 15
5 x- 15 = 5 x- 15
5 x- 15 = 51 x- 32
506 512 x+ 62 - 2 = 71 x+ 42
x= 0
3 x
3
=
0
3
3 x= 0
10 x+ 28 - 7 x- 28 = 7 x+ 28 - 7 x- 28
10 x+ 28 = 7 x+ 28
10 x+ 30 - 2 = 7 x+ 28
512 x+ 62 - 2 = 71 x+ 42
EXAMPLE 6
SECTION 2.1 Linear Equations in One Variable 53
Type of Linear Number of
Equation Solutions Indication when Solving
Conditional One Final line is number. (See Example 6(a).)
Identity Infinite; solution set Final line is true, such as (See Example 6(b).)
{all real numbers}
Contradiction None; solution set 0 Final line is false, such as - 15 =-20.(See Example 6(c).)0 =0.x=aNOW TRY
EXERCISE 6
Solve each equation. Decide
whether it is a conditional
equation,an identity,or a
contradiction.
(a)
(b)
(c) 10 x- 21 = 21 x- 52 + 8 x
- 312 x- 12 - 2 x= 3 +x
9 x- 31 x+ 42 = 61 x- 22NOW TRY ANSWERS
- (a) all real numbers ; identity
(b) ; conditional equation
(c) 0 ; contradiction
5065 6CAUTION A common error in solving an equation like that in Example 6(a)
is to think that the equation has no solution and write the solution set as. This
equation has one solution, the number 0, so it is a conditional equation with solution
set 506.
0
NOW TRY