Expression and Equation Examples (The Key Difference)

If you spend a few seconds scanning through any given math textbook, the odds are that you are going to find plenty of math expression and equation examples. Expressions and equations are two key words that are used very often in studies of algebra. But what exactly are math expressions, and how are they different from an algebraic equation?

Let’s dig into the main difference and look at some math expression and equation examples. I will also show you a few of my own personal tips and tricks so that you can start mastering these key algebra concepts!

What is an Expression and Equation?

What exactly are expressions and equations in math? Most math students (and even some teachers) use these important terms interchangeably. But there are a few key differences that you need to be aware of so that you can be successful in your study of algebra!

What are Equations?

I find it easier to first talk about what an equation is. This is mainly because you have probably seen a million examples of equations during your study of algebra.

An equation consists of a few key parts:

A variable (x for example)
An equal sign
A left-hand side
A right-hand side
Usually some sort of arithmetic operations (addition, subtraction, multiplication, or division)

Combining all of these together will produce an equation, or an algebraic sentence that conveys some sort of meaning.

Take the following equation for example:

$$ 3x – 2 = 4$$

This equation has left and right sides, a variable term (3x in this case), and most importantly, an equal sign separating the left side and the right side. There is also a subtraction operation between the first term and the second term on the left side.

When you read algebraic equations like this one using words, it becomes clear that there is a complete meaning. In this case, the equation reads: “two less than three times some number is equal to four.” It is a complete statement.

In the case of equations, there is only one one correct solution for the variable that can make the sentence true. Because we have something on the left side and the right side, the values of the variables in equations become fixed values.

We determine the unknown value using a little bit of algebra. Hopefully you can see that the value of the variable is x = 2 since this will make this sentence true on both sides of the equation! You can read more about how to solve algebraic equations below!

What are Expressions?

When it comes to understanding the difference between equations and expressions, it is helpful to look at a similar example as the one from above. Consider the following expression:

$$ 3x – 2 $$

Notice that this expression is almost the same as the equation you just looked at. However, there is one key difference: there is no equal sign!

Since there is no equal sign, we don’t have sides of the equation. Instead, the given expression is a single statement.

Unlike equations, notice that this mathematical statement has no complete meaning. As a word phrase, it simply says “two less than three times some number”. It doesn’t tell us what this is equal to.

Because of this, the value of x could be anything. It doesn’t have to be true on both sides of the equation. And depending what we choose x to be, we will have a different value of the expression.

What is the Difference Between an Expression and an Equation?

Before we get into examples of algebraic expressions and equations, watch this comparison of expressions and equations video for a quick comparison that explains the difference between mathematical expressions and equations!

As a tip, I find it helpful to remember that equation and equal sign both start with “equa-“. This is how you can remember that equations have an equal sign!

What are some examples of expressions and equations?

So we’ve talked about how an expression is just an algebraic statement, while algebraic equations are complete mathematical sentences. There are many different types of algebraic expressions and equations that range in complexity.

In algebra, we call equations that have a specific use formulas. You will encounter many different algebra formulas in your studies, particularly when working with real-world examples of algebra.

Let’s dig into a few different examples of expressions and equations in algebra so that you can start to deepen your understanding!

Examples of expressions with an exponent:

$x^2+4x+3$
$2^x-4$
$250(\frac{1}{2})^x$

Examples of equations with an exponent:

$x^2 +4x+3=0$
$2^x-4=4$
$10=250(\frac{1}{2})^x$

Examples of expressions with a square root:

$\sqrt{x}+2$
$\sqrt{2}+4$
$\sqrt{x}-2\sqrt{x}+1$

Examples of equations with a square root:

$\sqrt{x}+2=2$
$\sqrt{x}=0$
$-2x^2+\sqrt{x}=\sqrt{2x}$

Binomial expression examples:

$x – 4$
$3p + 5$
$-q + 7$

Examples of equations containing a binomial:

$x – 4 = -5$
$3p + 5=1$
$-q + 7=2$

Examples of expressions containing like terms:

$3x-2y+5x-y$
$-p-2p+p-3p$
$x^2-y^2+3x^2+4y^2$

Examples of equations containing like terms:

$3x+2=4x-1$
$5(x-2)=3(x+1)=0$

Polynomial Expression Examples:

$x^3+2x^2-4x+5$
$2x^4-2x-3$
$1+2x+3x^2$

Polynomial Equation Examples:

$x^3+2x^2-4x+5=0$
$2x^4-2x-3=-3$
$1+2x+3x^2=-1$

How to Simplify Expressions

When working with expressions, we know that we are not able to solve for the value of unknown variables since we do not have an equal sign. Instead, we can apply a series of tools to help us simplify a given expression. This includes:

Order of operations: applying the operation symbols in the correct order.
Collecting like terms: the process where any variables with the same degree are either subtracted or added together, while constant terms are also added or subtracted from one another.
Look for common factors: There may be times where you need to common factor an expression in order to make it simpler.

For example, simplifying the expression $x^2-y^2+3x^2+4y^2-x$ would result in $4x^2+3y^2-x$. Notice that the x term remains since it does not have the same base or exponent as the other terms.

Solving Algebraic Equations

When it comes to solving equations, we are able to apply algebraic techniques in order to determine the numerical value of the unknown variable.

To do this, we apply an opposite math operation to isolate and determine the value of the variable. Remember that equations are like a scale, where both sides must remain balanced!

So for example, since we have a minus sign in the equation $2x – 4 = -5$, the first step is to apply the opposite operation to each side of the equation: addition!

$$ \begin{split} 2x-4&=-5 \\ \\ 2x-4+4&=-5+4 \\ \\ 2x&=-1 \\ \end{split} $$

Notice that at this point we now have a single term of 2 times x on the left side of the equation. This tells us that we should apply the division operation to divide each side of the equation by 2 in order to solve for the value of the unknown variable.

$$ \begin{split} 2x&=-1 \\ \\ \frac{2x}{2}&=\frac{-1}{2} \\ \\ x&=-\frac{1}{2} \\ \end{split} $$

Therefore, the value of x in this equation is $\frac{1}{2}$.

Understanding Math Expression and Equation Examples

The expression and equation examples that I have shared here should help you understand the main difference between these important algebra concepts. And understanding the difference between expressions and equations in mathematics is an important skill!

Being able to use these terms correctly will help you communicate effectively with your teacher, other students, and mathematicians. It will also help you learn to simplify different types of algebraic expressions and solve algebraic equations!

I hope these expression and equation examples have helped you deepen your understanding and appreciation of algebra and algebra equations!

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