1.1: Definition of Variable
What if you had a jar filled with dimes and quarters? You know that the total of the coins in the jar is $8.60. How could you write an equation to represent this situation? After completing this Concept, you'll be able to use variables to write equations like this one with unknown quantities.
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CK12 Foundation: 0101S Language of Algebra
Guidance
No one likes doing the same problem over and over again—that’s why mathematicians invented algebra. Algebra takes the basic principles of math and makes them more general, so we can solve a problem once and then use that solution to solve a group of similar problems.
In arithmetic, you’ve dealt with numbers and their arithmetical operations (such as
For example, we might use the letter
Using variables offers advantages over solving each problem “from scratch.” With variables, we can:
 Formulate arithmetical laws such as
a+b=b+a for all real numbersa andb .  Refer to “unknown” numbers. For instance: find a number
x such that3x+1=10 .  Write more compactly about functional relationships such as, “If you sell
x tickets, then your profit will be3x−10 dollars, or “f(x)=3x−10 ,” where “f ” is the profit function, andx is the input (i.e. how many tickets you sell).
Example A
Write an algebraic equation for the perimeter and area of the rectangle below.
To find the perimeter, we add the lengths of all 4 sides. We can still do this even if we don’t know the side lengths in numbers, because we can use variables like
We are adding
It's customary in algebra to omit multiplication symbols whenever possible. For example,
Area is length multiplied by width. In algebraic terms we get:
Note:
In the above example, we found the simplest possible ways to express the perimeter and area of a rectangle when we don’t yet know what its length and width actually are. Now, when we encounter a rectangle whose dimensions we do know, we can simply substitute (or plug in) those values in the above equations. In this chapter, we will encounter many expressions that we can evaluate by plugging in values for the variables involved.
Example B
Eric has some money in his savings account. How much more money does he need in order to buy a game that costs $98?
Solution:
Let
Example C
Write an equation for the sum of 3 times some number and 5.
Solution:
Let
Watch this video for help with the Examples above.
CK12 Foundation: The Language of Algebra
Vocabulary
 We use symbols called variables (which are usually letters, such as
x, y, a, b, c, … ) to represent numbers and sometimes processes.

2l+2w by itself is an example of a variable expression;P=2l+2w is an example of an equation. The main difference between expressions and equations is the presence of an equals sign (=).
Guided Practice
Alex has a certain amount of nickels and dimes in a jar. Write an algebraic equation for how much money she has, in terms of how many nickles and dimes she has.
Solution:
Let
Since each dime is worth $0.10, the amount of money she has in dimes will be:
This means that the total amount of money
Simplifying the expressions, we get:
Practice
For 14, write the following in a more condensed form by leaving out a multiplication symbol.

2×11x 
1.35⋅y 
3×14 
14⋅z
For 510, write an equation for the following situations.
 The amount of money Andrea has in a jar full of quarters and dimes.
 The amount of money Michelle has in her coin purse if it only contains quarters, dimes and pennies.
 The sum of 7 and 6 times some number.
 4 less than 20 times some number.
 The amount of money you will earn if you are paid $10.25 an hour and spend $4.00 round trip to get too and from work.
 A father earns a $2000 dividend from an oil investment and distributes it equally amongst his children.
Equation
An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.substitute
In algebra, to substitute means to replace a variable or term with a specific value.Variable
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.Variable Expression
A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.Image Attributions
Here you'll learn how to represent unknown quantities with variables so you can write algebraic expressions and equations for realworld situations.
Concept Nodes:
Equation
An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.substitute
In algebra, to substitute means to replace a variable or term with a specific value.Variable
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.Variable Expression
A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.