Equality as Relationship Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Equality as Relationship.

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

Concept Recap

Understanding $=$ not as 'the answer is' but as expressing that two expressions represent the same value.

$3 + 2 = 5$ doesn't mean '3 + 2 makes 5'—it means they ARE the same.

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: The $=$ sign asserts two expressions name one identical value, not that an answer follows.

Common stuck point: The procedure for equality as relationship is the easy part; the trap is reading $=$ as 'the answer comes next'. Asking "Does the $=$ assert two expressions share one value (not just signal a result)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the $=$ assert two expressions share one value (not just signal a result)?

Worked Examples

Example 1

easy

Explain why

=

means 'the same value as', not just 'the answer is'. Use

3 + 5 = 4 \times 2

as an example.

Answer

Both sides equal 8;

=

expresses equivalence

First step

Left side:

3 + 5 = 8

Full solution

2
Right side: $4 \times 2 = 8$ .
3
Both sides equal 8, so the equation expresses a relationship between two equivalent expressions.
4
The $=$ sign means both sides name the same quantity.

Equality is a relationship between two expressions that have the same value. It is symmetric: if

a = b

then

b = a

Example 2

medium

a = b

and

b = c

, what can we conclude about

a

and

c

? Give a numeric example.

Example 3

medium

Explain why writing

4 + 5 = 9 + 3 = 12

is incorrect, then fix it.

Example 4

medium

Find the value of

\square

that makes

\square + 9 = 4 \times 6

true.

Example 5

medium

Are the equations

x + 3 = 7

and

2(x + 3) = 14

equivalent? Explain.

Example 6

hard

Find a value of

x

for which

x + 5 = 5 + x

is FALSE.

Example 7

hard

Show that the equation

2x + 5 = 2x + 1

has no solution.

Example 8

hard

Classify each as identity, conditional equation, or contradiction: (a)

x + 1 = x + 1

, (b)

x + 1 = 5

, (c)

x + 1 = x + 2

Example 9

challenge

Find all integers

x

and

y

such that

x^2 = y^2

Practice Problems

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

Example 1

easy

5 \times 6 = 30

the same kind of statement as

5 \times 6 = 3 \times 10

? Explain.

Example 2

medium

Given

2x + 1 = 9

, verify that

x = 4

is the solution. Then explain what equality means in this context.

Example 3

easy

True or false: in

3 + 4 = 7

, the

=

means '3 + 4 makes 7' as a command.

Example 4

easy

5 = 2 + 3

a valid equation?

Example 5

easy

Fill the blank to keep equality:

4 + 6 = 3 + \underline{\ \ }

Example 6

easy

Is the string

3 + 5 = 8 + 2 = 10

a correct use of

=

Example 7

easy

Does

7 = 7

express a true relationship?

Example 8

easy

Which expresses a relationship of sameness:

2 \times 3

2 \times 3 = 6

Example 9

easy

a = b

, is it also true that

b = a

Example 10

easy

Make it true:

10 - 3 = \underline{\ \ } + 2

Example 11

medium

A student writes

5 + 3 = 8 - 1 = 7

. Identify the false claim and rewrite correctly.

Example 12

medium

Find

x

so that

x + 4 = 4 + 9

using the meaning of equality.

Example 13

medium

\frac{6}{2} = 3

and

3 = \frac{9}{3}

, so

\frac{6}{2} = \frac{9}{3}

? Justify.

Example 14

medium

Which equations are TRUE: (i)

2+5 = 5+2

, (ii)

9-4 = 4-9

, (iii)

3\times2 = 2\times3

Example 15

medium

2x = 10

, explain why

2x

and

10

can be substituted for each other.

Example 16

medium

Make true with one number in the blank:

3 \times 4 = \underline{\ \ } \times 2

Example 17

medium

Explain why writing '

=

' between

x + 3

and

7

in '

x + 3 = 7

' is a claim, not a fact, until

x

is known.

Example 18

challenge

Use transitivity to show that if

a = b

and

b = c

and

c = 12

, then

a = 12

Example 19

challenge

Find all values of

x

making

x + 2 = x + 2

true, and explain what kind of statement this is.

Example 20

challenge

A student claims

\frac{1}{2} = 0.5 = 50\%

shows three different numbers. Use equality to explain why it is one number.

Example 21

medium

Make true:

8 + 5 = 20 - \underline{\ \ }

Example 22

medium

4 + 4 + 4 = 3 \times 4

a true equality? Justify.

Example 23

easy

True or false:

9 + 6 = 7 + 8

Example 24

easy

Which sign makes this true:

3 \times 4\ \square\ 6 + 6

? (

=

<

, or

>

)

Example 25

easy

True or false:

5 = 5 + 0

Example 26

easy

Make this true:

20 - 8 = 5 + \underline{\ \ }

Example 27

medium

Given

a = b

and

b = 7

, what is

a

? Which property did you use?

Example 28

medium

3(x + 4) = 3x + 12

a true equation for every value of

x

Example 29

medium

True or false: if

7 = x

, then

x = 7

Example 30

medium

5 + 2 \times 3 = 21

a true equation?

Example 31

hard

Find a value of

x

for which

3(x - 2) = 3x - 6

is FALSE.

Example 32

hard

For what value of

c

does the equation

4x + c = 4x + 9

have infinitely many solutions?

Example 33

hard

Find all values of

x

that satisfy

|x| = x

← Back to Equality as Relationship Formula Explained Practice Mode

Related Concepts

Equal Equations Equivalence

Background Knowledge

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

equal