PROPERTIES OF EQUALITY - MATHS - 2025

The properties of equality refer to the relationship between two mathematical objects, whether they are numbers or variables. It is denoted by the "=" symbol, which always goes between these two objects. This expression is used to establish that two mathematical objects represent the same object; in other word, that two objects are the same thing.

There are cases where it is trivial to use equality. For example, it is clear that 2 = 2. However, when it comes to variables it is no longer trivial and has specific uses. For example, if we have that y = x and on the other hand x = 7, we can conclude that y = 7 as well.

The example above is based on one of the properties of equality, as you will see shortly. These properties are essential to solve equations (equalities involving variables), which form a very important part of mathematics.

What are the properties of equality?

Reflective property

The reflexive property, in the case of equality, states that every number is equal to itself and is expressed as b = b for any real number b.

In the particular case of equality this property seems to be obvious, but in other types of relationships between numbers it is not. In other words, not every real number relationship meets this property. For example, such a case of the relation “less than” (<); no number is less than itself.

Symmetric property

The symmetric property for equality says that if a = b, then b = a. No matter what order is used in the variables, it will be preserved by the equality relation.

A certain analogy of this property with the commutative property can be observed in the case of addition. For example, due to this property it is equivalent to write y = 4 or 4 = y.

Transitive property

The transitive property on equality states that if a = b and b = c, then a = c. For example, 2 + 7 = 9 and 9 = 6 + 3; therefore, by the transitive property we have that 2 + 7 = 6 + 3.

A simple application is the following: suppose that Julian is 14 years old and that Mario is the same age as Rosa. If Rosa is the same age as Julián, how old is Mario?

Behind this scenario the transitive property is used twice. Mathematically it is interpreted as follows: let "a" be the age of Mario, "b" the age of Rosa and "c" the age of Julian. It is known that b = c and that c = 14.

By the transitive property we have that b = 14; that is, Rosa is 14 years old. Since a = b and b = 14, using the transitive property again we have that a = 14; that is, Mario's age is also 14 years old.

Uniform property

The uniform property is that if both sides of an equality are added or multiplied by the same amount, the equality is preserved. For example, if 2 = 2, then 2 + 3 = 2 + 3, which is clear, since 5 = 5. This property is most useful when trying to solve an equation.

For example, suppose you are asked to solve the equation x-2 = 1. It is convenient to remember that solving an equation consists of explicitly determining the variable (or variables) involved, based on a specific number or a previously specified variable.

Going back to the equation x-2 = 1, what you have to do is find explicitly how much x is worth. To do this, the variable must be cleared.

It has been erroneously taught that in this case, since the number 2 is negative, it passes to the other side of the equality with a positive sign. But it is not correct to say it that way.

Basically, what you are doing is applying the uniform property, as we will see below. The idea is to clear "x"; that is, leave it alone on one side of the equation. By convention it is usually left on the left side.

For this purpose, the number to "eliminate" is -2. The way to do it would be by adding 2, since -2 + 2 = 0 and x + 0 = 0. In order to do this without altering the equality, the same operation must be applied to the other side.

This allows us to realize the uniform property: since x-2 = 1, if the number 2 is added on both sides of the equality, the uniform property says that it is not altered. Then we have that x-2 + 2 = 1 + 2, which is equivalent to saying that x = 3. With this the equation would be solved.

Similarly, if you want to solve the equation (1/5) y-1 = 9, you can proceed using the uniform property as follows:

More generally, the following statements can be made:

- If ab = cb, then a = c.

- If xb = y, then x = y + b.

- If (1 / a) z = b, then z = a ×

- If (1 / c) a = (1 / c) b, then a = b.

Cancellation property

The cancellation property is a particular case of the uniform property, particularly considering the case of subtraction and division (which, basically, also correspond to addition and multiplication). This property treats this case separately.

For example, if 7 + 2 = 9, then 7 = 9-2. Or if 2y = 6, then y = 3 (dividing by two on both sides).

Analogously to the previous case, through the canceling property the following statements can be established:

- If a + b = c + b, then a = c.

- If x + b = y, then x = yb.

- If az = b, then z = b / a.

- If ca = cb, then a = b.

Substitution property

If we know the value of a mathematical object, the substitution property states that this value can be substituted in any equation or expression. For example, if b = 5 and a = bx, then substituting the value of "b" in the second equality we have that a = 5x.

Another example is the following: if "m" divides "n" and also "n" divides "m", then we must have that m = n.

Indeed, saying that "m" divides "n" (or equivalently, that "m" is a divisor of "n") means that the division m ÷ n is exact; that is, dividing "m" by "n" yields a whole number, not a decimal. This can be expressed by saying that there exists an integer "k" such that m = k × n.

Since "n" also divides "m", then there exists an integer "p" such that n = p × m. Due to the substitution property, we have that n = p × k × n, and for this to happen there are two possibilities: n = 0, in which case we would have the identity 0 = 0; op × k = 1, hence the identity n = n.

Suppose "n" is nonzero. Then necessarily p × k = 1; therefore, p = 1 and k = 1. Using the substitution property again, by substituting k = 1 in the equality m = k × n (or equivalently, p = 1 in n = p × m), we finally obtain that m = n, which was what we wanted to prove.

Power property in an equality

Just as previously it was seen that if an operation such as addition, multiplication, subtraction or division is done in both terms of an equality, it is preserved, in the same way other operations that do not alter an equality can be applied.

The key is to always perform it on both sides of the equality and make sure in advance that the operation can be performed. Such is the case of empowerment; that is, if both sides of an equation are raised to the same power, we still have an equality.

For example, since 3 = 3, so 3 ² = 3 ² (9 = 9). In general, given an integer "n", if x = y, then x ⁿ = y ⁿ.

Root property in an equality

This is a particular case of empowerment and it applies when the power is a non-integer rational number, such as ½, which represents the square root. This property states that if the same root is applied to both sides of an equality (whenever possible), the equality is preserved.

Unlike the previous case, care must be taken here with the parity of the root to be applied, since it is well known that the even root of a negative number is not well defined.

In the case that the radical is even, there is no problem. For example, if x ³ = -8, even though it is an equality, you cannot apply a square root to both sides, for example. However, if you can apply a cube root (which is even more convenient if you want to explicitly know the value of x), thus obtaining that x = -2.

References

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