Properties of Equality

The Central Idea

An equation states that two quantities are equal. If we transform that equation, we must preserve the equality.

Suppose

If we add the same quantity to both sides, subtract the same quantity from both sides, multiply both sides by the same quantity, or divide both sides by the same nonzero quantity, the resulting equation remains true.

These facts are not arbitrary classroom rules. They are consequences of the basic properties of equality.

The properties of equality provide the logical foundation for solving equations, rearranging formulas, substituting known values, and proving algebraic relationships.

Equality as a Balanced Relationship

A useful first model is a balance scale.

If two sides have equal weight, then adding the same weight to both sides preserves the balance. Removing the same weight from both sides also preserves the balance.

Algebra works similarly.

If

then subtracting \(3\) from both sides gives

so

The operation was applied equally to both sides, so the equality was preserved.

The balance model is useful, but the properties of equality extend far beyond physical balancing. They apply to numbers, expressions, functions, vectors, matrices, and many other mathematical objects whenever the relevant operations are defined.

Reflexive Property of Equality

Every mathematical object is equal to itself.

This is called the reflexive property of equality.

Examples include

and

The reflexive property may seem obvious, but it is fundamental. It expresses the idea that equality always relates an object to itself.

Symmetric Property of Equality

If one quantity equals another, then the equality can be read in the opposite direction.

If

then

This is the symmetric property of equality.

For example, if

then we may also write

Likewise, if

then

The order of the two sides does not change the equality being asserted.

Transitive Property of Equality

If two quantities are each equal to the same quantity, then they are equal to each other.

If

and

then

This is the transitive property of equality.

For example, if

and

then

The transitive property is one reason chains of equalities work:

Each link connects quantities that represent the same value.

Addition Property of Equality

If two quantities are equal, adding the same quantity to both sides preserves the equality.

If

then

This is the addition property of equality.

For example, from

we may add \(4\) to both sides:

Therefore,

The same quantity was added to both sides, so the original relationship was preserved.

Subtraction Property of Equality

If two quantities are equal, subtracting the same quantity from both sides preserves the equality.

If

then

For example, from

we may subtract \(6\) from both sides:

giving

The subtraction property is closely related to the addition property because subtracting \(c\) is equivalent to adding \(-c\).

Multiplication Property of Equality

If two quantities are equal, multiplying both sides by the same quantity preserves the equality.

If

then

For example, from

we may multiply both sides by \(5\):

so

Multiplication by zero also preserves truth:

However, multiplying by zero destroys information because both sides become \(0\). That matters when we want a transformation to be reversible.

Division Property of Equality

If two quantities are equal, dividing both sides by the same nonzero quantity preserves the equality.

If

and

then

For example, from

we may divide both sides by \(6\):

so

The condition \(c\ne0\) is essential because division by zero is undefined.

Substitution Property of Equality

If two quantities are equal, one may replace the other in an expression or equation wherever the replacement is valid.

If

then an occurrence of \(a\) may be replaced by \(b\), and vice versa.

Suppose

Then in

we may replace \(x\) with \(4\):

Likewise, if

then

The substitution property is the formal version of the replacement principle introduced in the previous lesson.

A Summary of the Core Properties

Properties Versus Operations

It is useful to distinguish the operation from the justification.

Consider

We perform the operation of subtracting \(9\):

The reason this is legitimate is the subtraction property of equality.

Similarly, in

we perform division by \(4\):

The justification is the division property of equality.

This distinction becomes important in formal reasoning. A line of algebra shows what was done; the relevant property explains why it was allowed.

Undoing Operations

Many equation-solving steps are designed to undo operations.

Suppose

We first subtract \(5\) from both sides:

Then divide both sides by \(3\):

The subtraction undoes the addition of \(5\), and the division undoes the multiplication by \(3\).

This works because inverse operations combine with the properties of equality. The operation removes part of the expression, while the property guarantees that doing the same thing to both sides preserves equality.

Working with More Complicated Expressions

The quantities added, subtracted, multiplied, or divided do not have to be simple numbers.

If

then we may add an entire expression \(f(x)\) to both sides:

Likewise, we may multiply both sides by another expression:

However, if we later divide by \(g(x)\), we must know that

This is one place where domain restrictions enter algebraic manipulation.

Why the Same Operation Must Be Applied to Both Sides

Suppose

If we subtract \(2\) only from the left side, we would get

which is not equivalent to the original equation.

The original equation has solution

while the altered equation has solution

The equality was not preserved because the transformation changed one side without making the corresponding change to the other.

Equation solving is therefore not about “moving” terms across an equal sign by magic. What appears to be movement is usually shorthand for applying the same legitimate operation to both sides.

What “Move It to the Other Side” Really Means

Students often hear instructions such as “move the \(5\) to the other side and change its sign.” This shortcut can be convenient, but it hides the underlying operation.

From

the mathematically explicit step is

Then

The \(5\) did not literally travel across the equal sign. We subtracted \(5\) from both sides using the subtraction property of equality.

Understanding the underlying property makes the shortcut safer and helps prevent sign errors.

Properties of Equality and Functions

The same idea can be expressed more generally using functions.

If

then applying the same well-defined function \(f\) to both sides gives

For example, if

then

This preserves truth: equal inputs produce equal outputs.

However, the reverse implication may fail. From

we cannot always conclude

because \(a\) and \(b\) could be opposites.

For example,

but

This distinction will matter later when we study reversible steps and solution preservation.

Preserving Truth Is Not Always the Same as Preserving Equivalence

Suppose

Squaring both sides gives

The new equation is true for every solution of the original equation, so the transformation preserved truth in the forward direction.

But the new equation has an additional solution:

Therefore the two equations are not equivalent.

This is an important refinement:

• A transformation may preserve every original solution.
• Yet it may still introduce new solutions.
• To preserve full equivalence, the transformation must preserve the entire solution set.

The elementary properties of equality involving addition, subtraction, multiplication by a nonzero quantity, and division by a nonzero quantity are especially useful because they can be reversed under the same conditions.

Multiplication by Zero: A Special Case

The multiplication property allows us to multiply both sides of a true equation by zero.

If

then

so

The resulting equality is true, but it no longer tells us what \(a\) and \(b\) were.

For example,

implies

which becomes

The new equation is satisfied by every real number, so it is not equivalent to the original equation.

This is a useful reminder that a valid forward operation need not be reversible.

Division Requires Special Care

Division can be especially dangerous when the divisor contains a variable.

Suppose

If we divide both sides by \(x\), we obtain

and therefore

But the original equation also has the solution

Dividing by \(x\) silently assumed that

That assumption discarded a legitimate solution.

Whenever dividing by a variable expression, ask whether that expression could equal zero.

Worked Example: Solving with Explicit Justification

Solve

Add \(7\) to both sides:

By the addition property of equality,

Divide both sides by \(5\):

By the division property of equality,

The equation has been transformed step by step without changing its solution set.

Worked Example: Rearranging a Formula

Suppose

where \(A\) is area, \(l\) is length, and \(w\) is width.

To solve for \(w\), divide both sides by \(l\), provided \(l\ne0\):

Therefore,

The same properties used in elementary equations also justify rearranging formulas with several variables.

Worked Example: Substitution and Transitivity

Suppose

and

By substitution, replace \(b\) with \(5\):

Therefore,

If we separately establish

then the transitive property links the equalities:

Much routine algebra uses several properties of equality at once.

Practice

1. Name the Property

If

then

Which property justifies this?

2. Solve and Justify

Solve

3. Identify the Restriction

From

under what condition may we divide both sides by \(a\) and conclude

4. Is the Transformation Reversible?

Starting with

we cube both sides:

Over the real numbers, does this transformation preserve equivalence?

5. Find the Lost Solution

Consider

A student divides both sides by \(x\) and obtains \(x=-4\). What solution was lost?

Key Ideas

• The properties of equality justify many standard algebraic transformations.
• Reflexive, symmetric, and transitive properties describe the basic structure of equality itself.
• Addition and subtraction of the same quantity on both sides preserve equality.
• Multiplication by the same quantity preserves equality, but multiplication by zero may destroy information.
• Division by the same quantity preserves equality only when the divisor is nonzero.
• Substitution allows one equal quantity or expression to replace another.
• A valid forward transformation does not automatically guarantee full equivalence.
• Variable divisors require special attention because they may equal zero for some possible solutions.
• Formal properties explain the logic behind familiar shortcuts such as “moving a term to the other side.”

Where This Leads

The properties of equality explain why equations can be transformed without arbitrarily changing their meaning.

Next, we will apply these ideas to Combining Like Terms, where equivalent pieces of an expression are collected into a more compact form while preserving the value of the expression.