Equations and Identities: Difference & Maths

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Expressions, equations, identities, and formulae

This section will consist of lots of definitions and examples. However, it is important that you understand each of the key terms as you may be given a particular example in your GCSE exam and asked to determine whether it is an expression, equation, identity, or formula. Here we go...

Definition of expressions

An expression is a collection of mathematical terms, related by mathematical operations. A mathematical term is a single mathematical Number or letter, for example, x or 3. We could also have 3x², where 3 is known as the coefficient of x.

The following are examples of mathematical expressions:

$5 x + 1$
$3 x^{2} - 9 x + 1$
$7 x + 2 y + z$
$a x^{2} + 1$
$6 x^{8} + 7 y^{5} + 2 z^{3} - 1$
$7 k$

Definition of equations

Once we have a firm grasp on what mathematical expressions are we can start to build connections between them, in order to compare them.

If we break down the word equation we get ‘equa-tion’. Now ’equa’ sounds very similar to ‘equal’, which is no coincidence: an equation is a statement that two mathematical expressions shall be the same.

An equation is a statement that two mathematical expressions shall be the same.

An equation is expressed with an equal sign $=$ between two mathematical expressions. The fancy word for the equal sign is the equality symbol.

In a few words, anything with an equal sign is an equation. Sounds simple, right? Here are some examples of equations:

$x = 2$
$a + b = c$
$x^{2} + y = 2$
$y - 2 = 5 y + 1$
$a^{2} - 5 a - 2 = 0$
$e^{πi} + 1 = 0$

Solution of an equation

It is important to note here that two expressions may only be equal under specific conditions. For example, if we are told $x - 2 = 0$ , we know this is an equation because there is an equality symbol. However, using the basic rules of addition, we know that the equation can only be true if $x = 2$ . This is called the solution of the equation.

The solution of an equation is the set of all values that, when substituted for the variables in the equation, make the equation true.

Definition of identities

Some expressions are always equal to each other, regardless of the values of the variables they contain. In this case one speaks of a mathematical identity:

A mathematical identity is where two mathematical expressions are always identical. An identity is expressed using the identity symbol $\equiv$ , which looks a bit like an equals sign with an extra line.

$2 x + 4 x \equiv 6 x$
$2 a^{2} + 3 a^{2} \equiv 5 a^{2}$
$11 x - 3 x \equiv 8 x$
$2 y + 3 y \equiv 5 y$
$9 p + p \equiv 10 p$

Differences between identities and equations

Often, one of the biggest challenges is determining whether something is an equation or identity. In this section, we will discuss how to establish whether something is an equation or an identity and note the key difference between them.

As established, an equation shows that two expressions are equal. However, they may only be equal for a specific value. For example, if we have $2 x + 1 = 5$ , this equation is only true when $x = 2$ . Identities on the other hand show that two expressions are always identical. For example, we could say that $6 x^{2} + 2 x^{2} \equiv 8 x^{2}$ , since, no matter what the value of $x$ is, the two expressions are always the same.

If we asserted the identity $a x^{2} + b x \equiv 2 x^{2} + 3 x$ , we know that the left-hand side is equal to the right-hand side for all values of $x$ because we have an identity symbol. However, the only way for this to be possible is if $a = 2$ and $b = 3$ . In this case, we would have $2 x^{2} + 3 x \equiv 2 x^{2} + 3 x$ which we know is true for all values of $x$ .

We could say that since identities show equality between expressions, all identities are equations. However, not all equations are identities so it is important that you are aware of the key difference between the two.

Here is again the main point: Equations show equality under at least one condition, identities show equality under all conditions.

Definition of formulae

We have a fourth thing to consider, and that is a formula. While we are here, it is worth mentioning that the plural of formula is formulae. Now, onto some more definitions and examples...

A formula is a special type of equation representing a general fact or rule one can work with.

A mathematical formula is a type of equation as all formulae have an equal sign. However, they are equations with a specific purpose. They give us a way of working something out. For example, if we wanted to convert degrees Fahrenheit to degrees Celsius, we could use a formula. There are many formulae that are specifically useful for GCSE mathematics, including the quadratic formula, the trigonometric formulae, and also the speed, distance, and time formula. In the example below we will look at some specific formulae.

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

This is quite possibly one of the most iconic formulae in GCSE mathematics – the quadratic formula. This article is not about the quadratic formula specifically, so we will not talk about it in too much depth. However, this is just a friendly reminder that you probably should learn it.

$m a s s = v o l u m e \times d e n s i t y$

Again, this article is not specifically about volume, mass, and density so we do not need to talk about it in too much depth. However, you do need to learn this formula at some point! It means that if you have the density of an object and volume, you can work out the mass. It is very handy.

$s p e e d = \frac{d i s \tan c e}{t i m e}$

This is another classic. Speed, distance, and time. You do need to know this formula, not just for GCSE mathematics, but for physics too. But once you know it, you can work out the speed of any moving object given the distance and time.

$a^{2} + b^{2} = c^{2}$

This is Pythagoras's Theorem. However, by definition, it is also a formula, as it enables us to work out an unknown quantity which in this case, is the missing side of a right-angled triangle. You need to know this for your GCSE exams.

$v = u + a t$

This is a formula that relates the final velocity of a moving object, with the initial velocity, time, and acceleration. It is one of the SUVAT formulae which you may come across if you study A-level Mathematics. You don't specifically need to know this formula, you just need to be able to know that it is a formula, as it enables us to work out something specific (eg, the initial velocity, final velocity, acceleration, or time)

$E = m c^{2}$

You may have seen this formula before. It is one of Einstein's most famous formulae relating mass to energy and the speed of light. It is quite famous, but you do not need to know this formula for GCSE mathematics. You simply need to know that it is a formula as it enables us to work something out.

Above are just a few examples of some formulae that you may or may not need to know for your GCSE maths exams. There are others, however, this article is not about going over every single one. Instead, it is about being able to see a formula, and subsequently being able to state that it is a formula, as opposed to an equation, identity, or expression.

Below we will cover some relevant examples to this topic. You will need to know the differences between an expression, identity, equation, and formula so let's just quickly recap the differences between these four things.

An expression is a collection of mathematical terms.
An equation is anything with an equal sign, hence 'equa' 'tion'.
An identity is where two mathematical expressions are identical, denoted with an identity symbol, $\equiv$ .
A formula is a type of equation where we are working out something specific, for example, the mass of an object.

Examples of identities and equations

Now that we have thoroughly defined some mathematical terms, we will go through some questions that you may come across in your GCSE exam.

Label the following either an identity, equation, expression, or formula:

$A = {πr}^{2}$
$9 x + 8$
$x^{2} + 4 x + 3 = 0$
$4 x + x = 5 x$

Solution:

1. is the formula for the area of a circle. It enables us to work out the area given the radius. Thus the first one is a formula.

2. has no equality sign, and is simply a collection of mathematical terms connected with an addition symbol. Thus it is an expression.

3. has an equals sign, and thus it is an equation. It is only true for specific values of $x$ , ( $x = - 3, x = - 1$ ), thus it is not an identity.

4. has an equals sign and thus is an equation. However, this equation is true for all values of $x$ and so it is an identity and can be expressed using the identity symbol $\equiv$ .

For the below identity, work out the values of $a$ and $b$ :

$a x^{2} + b x^{2} + a x \equiv 5 x^{2} + 3 x$

Solution:

We know that it is an identity, and so the left-hand side must be equal to the right-hand side. The coefficient of $x$ on the right-hand side is $3$ and so the coefficient on the left-hand side must also be $3$ . Thus, $a = 3$ .

On the left-hand side, we could group up the coefficients of $x^{2}$ as follows:

$(a + b) x^{2} + 3 x \equiv 5 x^{2} + 3 x$ . Therefore, we could say that $a + b = 5$ since the coefficient of $x^{2}$ on both sides must be the same. Since we already know that $a = 3$ , we can say that $3 + b = 5$ and so $b = 2$ . Thus, $a = 3$ and $b = 2$ .

Equations and Identities - Key takeaways

An expression is a collection of mathematical terms, related by mathematical operations.
An equation is any mathematical relation expressed with an equal sign.
An identity is where two mathematical expressions are always identical.
A formula is an equation that enables us to work out something specific.
Identities are expressed using the symbol $\equiv$ which is like an equal sign with an extra line.
The difference between an equation and an identity is that equations state equality under a specific condition, identities show that two expressions are always equal.

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Frequently Asked Questions about Equations and Identities

What is the difference between an equation and an identity?

An equation is true for certain values, an identity is true for all values.

What type of equations are identities?

Identities are equations where two expressions are always equal, as oppose to just under certain conditions.

What is an expression, equation and identity and formula?

An expression is a collection of mathematical terms. An equation is where two expressions are equal for some values. An identity is where two expressions are always equal, regardless of the values. A formula is a rule for working something specific out.

What is an equation in math?

An equation is any mathematical relation expressed with an equality sign.

What is an identity in math?

An identity is where two expressions are always true.

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Equations and Identities

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