Writing Expressions and Equations

Learning Objective

• Translate word problems into algebraic expressions and equations.

Learning algebra usually starts with the tools and methods of solving equations. But it’s just as important to know how to write equations and expressions. Once this course is over, most of us won’t go around looking up equations to solve. But all of us will encounter questions in our lives that can be answered if we know how to turn a situation into an equation. How much do I have to save from every paycheck to cover my yearly car insurance? How do I cut up this pan of brownies so that everyone gets one? When do I have to get up so that I can get dressed, have breakfast, and still get to class on time? Algebra can help us solve problems like these.

For algebra to be effective in answering real-world questions, we must be able to write equations that accurately describe the situations we want to explore. Equations need to describe both what we know and what we want to find out. The quantities that we already know are written in an expression as numbers. Because they are numbers and because they don’t change, these known quantities are called numeric constantsA quantity that has a known, fixed value..

The values that we don’t know or that could change are written as symbols. Because they vary, these are called variablesA symbol that represents an unknown value..

The relationships among variables and constants are described by mathematical symbols that show operationsA mathematical procedure, such as addition, subtraction, multiplication, and division. to be carried out, such as `+` and `-`.

Once we’ve identified the constants, variables, and operations that describe what we know and what we want to know and how they are related, we can put them all together into an expression or equation.

Let’s look at an example of how we can turn a real-life situation into an algebraic equation.

The drama club at a high school is going to raise money by printing calendars that feature photos of scenes from its recent plays. The cost of printing the calendars is `$5.50` per calendar. The photographer also charges a one-time cost of `$200` for taking the photos. The club has `$1,500` to cover the initial costs of the calendar. How can we help them decide how many calendars they can order? By writing an equation, of course. Let’s start by identifying the knowns, the unknowns, and the relationships between them.

In this situation, there are three known, constant values:

• the cost per calendar, which is `$5.50`
• the cost of taking the photographs, which is `$200`
• the amount of money that can be spent, which is `$1,500`

There is one unknown variable, which is quantity:

• the number of calendars the club can print, which we will call `n`

The relationships among the terms includes two operations:

• multiplying the number of calendars times the cost per calendar will give the printing costs
• adding the photography cost to the printing cost will give the total cost

The total cost must equal the amount they have to spend.

Now we can put the constants, the variables, and the operations together into an equation that combines the costs and sets them equal to the money available. That equation is `5.5n + 200 = 1,500`. All the club has to do now is solve this equation for `n`, and they’ll know how many calendars they can afford to order.

It’s very useful to know how to convert word problems and situations into algebraic equations. When you write an equation, begin by asking yourself these questions:

Which quantities are known and constant? These will be represented in the equation with numbers.

Which quantities are unknown and changeable? These will be represented in the equation with variables.

What is the relationship between constants and variables? These will be represented in the equation with operations.

Once you identify what you know and what you want to find out, you can build an equation that will let you solve the problem.