expression: a collection of constants, variables, and math symbols that expresses a valueequation: two expressions separated by an equal sign
Step-by-step explanation:
Expressions
The very simplest expression is simply a constant, such as 0, or 1 or 0.6 or -37.2.
An expression may also be a variable, often a single letter, but not restricted to that. A variable can be any set of symbols recognizable as being a single unit with an unambiguous meaning, such as This_is_a$variable$, or <the meaning of life>. Generally, a variable will have a particular meaning in the context of a problem, so an attempt is often made to have it have mnemonic value. For example, A may be used to represent area, P may be used to represent perimeter.
At the next level, an expression is a collection of constants, variables, and math symbols or operators.
Each math symbol has its own meaning. Some operate on a single number or variable or expression, some operate on more than one number or variable or expression. Some are "prefix" operators, such as "-" in -3. Some are "postfix" operators, such as "!" in n!. Others require two operands, one on either side, such as "+" in a+4. Others are multi-part symbols with parts that have different meanings. (The Σ symbol, below, is one of those.) Some symbols act to group parts of an expression, or to both group things and perform an operation on that group. Some have multiple purposes, such as "-" in the above example and in b-5. Some operators are implicit or invisible. For example, the two variables x and y can be indicated as being multiplied simply by writing them next to each other as xy. (Note that "xy" could also be the (poorly-chosen) name of a single variable.) This form is commonly used when a variable is multiplied by a constant: 2x.
Expressions can get somewhat complicated. For example, here's one that uses a variety of math symbols:
 
The rules for interpreting an expression are embodied in the "order of operations". By following these rules, anyone who reads an expression will see it have the same value.
Different parts of an expression have different names, as do the operands for different math operators. It is helpful to learn the vocabulary so that one can understand what is being talked about in a discussion of expressions and their parts.
In short, an expression is a collection of math symbols and the constants or variables or expressions they operate on.
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Equations
An equation is two expressions separated by an equal sign (=). A simple equation is ...
 x = 1
The equal sign means the two expressions have the same value. There are rules regarding the kinds of transformations that can be done on an equation so that the equal sign remains valid. For example, we can multiply both sides of the equation by 2:
 2x = 2
Often times, the purpose of writing an equation is to express a relationship. For example, we might write ...
 Area = Length × Width
If we use single-letter variables for the concepts of area, length, and width, then we could write the same equation as ...
 A = LW
where the multiplication is indicated by simply writing the variables next to each other.
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"Solving" an equation generally means finding the value(s) of the variable(s) such that the equation is a true statement. The above equation in x, for example, is true when the value of x is 1. Putting that value of x into the equation makes it become ...
 1 = 1 . . . . a true statement
For any other value of x, the equation is not true. For example, ...
 0 = 1 . . . . a false statement
