- Topics At a Glance
- Variables
- Variables as Unknown Quantities
- Variable Notations
- Constants
- Expressions and Equations
- Rearranging Expressions
- Commutative Properties
- Associative Properties
- Distributive Properties
- Factoring (Distributive Property in Reverse)
- Combining Like Terms
- Eliminating Parentheses
- Simplifying
**Equations, Functions, and Formulas**- Equations
- Functions
- Independent and Dependent Variables
**Formulas**- Applications to Toolbox
- Evaluating Expressions by Substitution
- Evaluating Formulas by Substitution
- Geometric Formulas
- Four-Sided Shapes
- Three-Sided Shapes
- Circles
- Unit Conversion
- Temperatures
- Weights
- Distances and Speeds
- Money
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

It is been a long time since you have needed to deal with formula on a daily basis. You probably transitioned to solid foods at around five months. But, now it is back.

All right, this is a different type of formula. A **formula** in the mathematical sense is the special case of a function where we can write an equation expressing the dependent variable as an expression involving the independent variables. In fact, in the previous section, the equations we got describing dependent variables in terms of another variable were examples of formulas. And, one-third less powdery than the leading brand!

1. Draw a rectangle with side lengths *l* and *w*. On second thought, save yourself the trouble—we will draw one for you. We're quite good at rectangles.

Let *A* represent the area of the rectangle. Then the equation *A* = *lw* is a formula describing *A* in terms of *l* and *w*.

**Be Careful:** Whenever we have an equation that has one variable all by itself on one side of the = sign, and on the other side has an expression involving one or more different variables, we consider this equation to be a **formula**. We think of the variable that is all by itself as the **dependent variable**, and any variables in the expression on the other side of the equality as **independent variables**. Gosh, we had no idea how much fun it was to make words **bold**. This could **really** start to become **addictive**.

2. Let *y* = *x*^{2}. This formula says that, to find the value of *y* from the value of *x*, we square the value of* x*.

3. In the formula *p* = 10 · *h*, *p* is the dependent variable and *h* is the independent variable.

4. In the formula *A* = *lw*, *A* is the dependent variable while *l* and *w* are independent variables.

5. In the formula *y* = *x*^{2}, *y* is the dependent variable and *x* is an independent variable. It finally has its own car and a small apartment in Midtown.

There are some geometric formulas (link) that appear frequently in algebra. It is a good idea to become familiar with them. You may want to think about inviting them to lunch, your treat. Get to know them, ask about their families. You know. The yooj.

There are also some unit conversion formulas (link) that you usually won't be expected to know off the top of your head, but which may actually be useful sometime. These guys you only need to take to breakfast.

**Be Careful:** Always know what your variables mean. Do whatever it takes. Seriously. Write out in the margin of your paper, "*H* stands for the area of half a circle" or something like that. It is a lot easier to answer questions correctly if you know the meanings of all the symbols you are using. Just as it is easier to avoid getting into a car accident when you know that that red octagonal sign means "stop."

Example 1

Jim gets paid $10 per hour. The amount Jim gets paid in a week depends on the number of hours he works that week. How should we represent each of these quantities? |

Exercise 1

The formula for the area of a circle is *A* = π*r*^{2}. What do the letters *A* and *r* represent?

Exercise 2

The formula for the area of a circle is *A *= π*r*^{2}. Which is the dependent and which is the independent variable?

Exercise 3

The formula for the area of a circle is *A* = π*r*^{2}. Let *H* be the area of half a circle. Write a formula describing *H* in terms of *r*.