# The Basics of Evaluating Equations

An *equation** *is a mathematical statement that tells you that two things have the same value — in other words, it’s a statement with an equals sign. The equation is one of the most important concepts in mathematics because it allows you to boil down a bunch of complicated information into a single number.

Mathematical equations come in a lot of varieties: arithmetic equations, algebraic equations, differential equations, partial differential equations, Diophantine equations, and many more.

*A**rithmetic equations* are equations involving numbers, the Big Four operations, and the other basic operations (absolute values, exponents, and roots). Here are a few examples of simple arithmetic equations:

And here are a few examples of more-complicated arithmetic equations:

## Hey, it’s just an expression

An *expression* is any string of mathematical symbols that can be placed on one side of an equation. Mathematical expressions, just like equations, come in a lot of varieties. *A**rithmetic expressions* contain numbers, the Big Four operations, and a few other basic operations. Here are a few examples of simple expressions:

And here are a few examples of more-complicated expressions:

## Evaluating the situation

At the root of the word *evaluation* is the word *value.* In other words, when you evaluate something, you find its value. Evaluating an expression is also referred to as *simplifying, solving,* or *finding the value of an expression.* The words may change, but the idea is the same — boiling down a string of numbers and math symbols to a single number.

When you evaluate an arithmetic expression, you simplify it to a single numerical value — in other words, you find the number that it’s equal to. For example, evaluate the following arithmetic expression:

7 x 5

How? Simplify it to a single number:

35

## Putting the three E’s — equations, expressions, and evaluation — together

You’re probably dying to know how the Three E’s — equations, expressions, and evaluation — are all connected. *Evaluation** *allows you to take an *expression* containing more than one number and reduce it to a single number. Then you can make an *equation,* using an equals sign to connect the expression and the number. For example, here’s an *expression** *containing four numbers:

1 + 2 + 3 + 4

When you *evaluate** *it, you reduce it to a single number:

10

And now you can make an *equation* by connecting the expression and the number with an equals sign:

1 + 2 + 3 + 4 = 10