Identity vs Equation Examples: 46 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Identity vs Equation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An identity is an equation that holds true for all possible values of the variable, such as $(a+b)^2 = a^2 + 2ab + b^2$ . A conditional equation is true only for specific values, like $x + 3 = 7$ (true only when $x = 4$ ).

$a + a = 2a$ is always true (identity). $x + 3 = 7$ is only true when $x = 4$ (equation).

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An identity holds for every value; a conditional equation holds only for special values.

Common stuck point: The procedure for identity vs equation is the easy part; the trap is calling every equation an identity. Asking "Does the equality hold for EVERY value of the variable, or only for special ones?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the equality hold for EVERY value of the variable, or only for special ones?

Worked Examples

Example 1

easy

2(x + 3) = 2x + 6

an identity or an equation with specific solutions?

Answer

Identity (true for all

x

First step

Expand the left side:

2x + 6

Full solution

2
Compare: $2x + 6 = 2x + 6$ is always true, regardless of $x$ .
3
This is an identity—it holds for all values of $x$ .

An identity is true for every value of the variable. An equation with specific solutions is true only for certain values. If simplifying both sides gives the same expression, it is an identity.

Example 2

medium

Classify: (a)

x^2 - 1 = (x+1)(x-1)

and (b)

x^2 - 1 = 0

Example 3

medium

(x+1)^2=x^2+1

an identity? If not, find the solution set.

Example 4

medium

Show that

a^2-b^2=(a-b)(a+b)

is an identity.

Example 5

hard

Find all

k

for which

(x+k)^2=x^2+4x+k^2

is an identity.

Example 6

hard

Decide for which value of

k

the equation

\frac{x+k}{x-2}=1+\frac{4}{x-2}

is an identity (on its domain).

Example 7

challenge

Find all

a

for which

(x+a)^3=x^3+3ax^2+3a^2x+a^3

is an identity in

x

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

5x = 20

an identity or an equation?

Example 2

medium

a(b + c) = ab + ac

an identity?

Example 3

easy

x+x=2x

an identity or a conditional equation?

Example 4

easy

x+3=7

an identity or a conditional equation?

Example 5

easy

Classify:

(a+b)^2=a^2+2ab+b^2

Example 6

easy

Classify:

2x=10

Example 7

easy

Classify:

3(x+2)=3x+6

Example 8

easy

Classify:

x^2=9

Example 9

easy

Classify:

a\cdot 0 = 0

Example 10

easy

Classify:

5x-1=2x+8

Example 11

medium

Determine whether

\frac{x^2-1}{x-1}=x+1

is an identity, and state any restriction.

Example 12

medium

(x-1)^2=x^2-1

an identity? If not, find all

x

where it holds.

Example 13

medium

Solve

4x+2=2(2x+1)

and interpret the result.

Example 14

medium

Solve

3x+5=3x+2

and interpret the result.

Example 15

medium

Decide if

\sin

-free statement

2(x+3)=2x+5

is an identity, conditional, or contradiction.

Example 16

medium

For what value of

k

does

2x+k=2x+7

become an identity?

Example 17

challenge

Find all values of

m

for which

mx+4=2x+4

has infinitely many solutions, exactly one, or none.

Example 18

challenge

Prove that

(a+b)^2-(a-b)^2=4ab

is an identity.

Example 19

challenge

For which value(s) of

c

does

x^2+c=(x+1)(x-1)+1

hold for all

x

Example 20

medium

Solve

3(x+1)=3x+3

and classify the result.

Example 21

medium

\frac{2x}{2}=x

an identity? State any restriction.

Example 22

medium

For what value of

b

5x-b=5x-3

an identity?

Example 23

easy

Classify:

4(x+2)=4x+8

Example 24

easy

Classify:

x-7=0

Example 25

easy

Classify:

7x=7x

Example 26

easy

Classify:

x+5=x+3

Example 27

easy

Classify:

x^2=4

Example 28

easy

Classify:

0\cdot x=0

Example 29

medium

Solve

2(x-3)=2x-6

. Classify the result.

Example 30

medium

Solve

6x+2=3(2x+1)

. Classify the result.

Example 31

medium

Classify:

\frac{x^2-9}{x-3}=x+3

Example 32

medium

Find

k

so that

3x+k=3x-2

is an identity.

Example 33

medium

Find all

x

for which

\frac{x}{x}=1

Example 34

medium

Classify:

|x|=x

Example 35

medium

Find

a

and

b

a(x+1)+b(x-1)=4x+2

is an identity.

Example 36

hard

Classify

\frac{1}{x-1}+\frac{1}{x+1}=\frac{2x}{x^2-1}

. State any restrictions.

Example 37

hard

Solve

mx+5=4x+5

for

m

so that all real

x

are solutions.

Example 38

hard

\sin^2 x+\cos^2 x=1

an identity?

Example 39

challenge

Find all pairs

(a,b)

for which

a(x-1)+b(x+2)=3x+1

is an identity.

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Related Concepts

Equations Algebraic Identities Solving Linear Equations

Background Knowledge

These ideas may be useful before you work through the harder examples.

equations