1. An expression is a "phrase" that represents a number.
2. An equation sets two expressions equal to each other.

A variable is a symbol used to represent a number. In the expression 8x + 17, x is the variable. In the equation = 25, y is the variable. Variables need not be letters; for example, the expression 6 + 2 contains the variable .

An expression or equation may contain more than one variable. The expression 7 - 3r + 12t contains the variables r and t, and the equation 6y = 24x contains the variables x and y.

An algebraic expression is an expression that contains one or more variables.
An algebraic equation is an equation that contains one or more variables.

Note: When we write two quantities next to each other, or next to each other separated by parenthesis, the multiplication sign is implied. For example, 4r means 4×r and 3(4 + 2) means 3×(4 + 2).

We can represent word statements by algebraic expressions and equations. For example, "the number of apples in the bucket plus 5 more apples" can be represented by the expression a + 5. The statement "twice the number of miles I ran is equal to 12" can be represented by the equation 2m = 12. Here are the steps to writing a word statement as an expression or an equation:

1. Identify the unknown quantity (or quantities).
2. Choose variable(s) to represent the unknown(s).
3. Identify the operations on the variables.

If a phrase asserts that two quantities are equal, then it is mathematically expressed by an equation. To determine this equation's component expressions, separate the two quantities and follow the above steps for each.

Observe the steps for the expression "the number of in the bucket plus 5 more apples":

1. >The number of apples is unknown.
2. Choose a = the number of apples.
3. The number of apples plus 5 more: a + 5.

Thus, the statement can be represented by the expression a + 5.

Observe the steps for the equation "twice the number of miles I ran is equal to 12":
Separate the quantities into "twice the number of miles I ran" and "12".
The left-hand side of the equation:

1. The number of miles I ran is unknown.
2. Choose m = the number of miles I ran
3. Twice the number of miles I ran: 2m

The right-hand side of the equation:

1. There are no unknowns.
2. Since there are no unknowns, there are no variables.
3. The only "operation" is the number 12.

Thus, the statement can be represented by the equation 2m = 12.

Here is an example of a word statement with more than one unknown--this translates into an expression with more than one variable:
"The height of the rectangle plus the width of the rectangle, all doubled."

1. The height of the rectangle and the width of the rectangle are unknown.
2. Choose h = height of rectangle and w = width of rectangle.
3. The height of the rectangle plus the width of the rectangle, all doubled: (h + w) x 2--we can also write this as 2(h + w)

Thus, the statement can be represented by the expression 2(h + w).

Here is an example of a word statement that translates into an equation with variables on both sides:
"Dan's height minus 1 foot, all multiplied by 2, is equal to Heather's height plus Dan's height."
Separate the quantities into "Dan's height minus 1 foot, all multiplied by 2" and "Heather's height plus Dan's height."
The left-hand side of the equation:

1. Dan's height is unknown.
2. Choose d = Dan's height in feet
3. Dan's height minus 1 foot, all multiplied by 2: 2(d - 1)

The right-hand side of the equation:

1. Heather's height and Dan's height are unknown.
2. Choose h = Heather's height in feet. We have already chosen d = Dan's height in feet
3. Heather's height plus Dan's height: h + d

Thus, the statement can be represented by the equation 2(d - 1) = h + d.

As we saw in step two of the previous problem, if we choose a variable to represent an unknown quantity on one side of an equation, we must use the same variable to represent the same quantity on the other side.