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Determine an Equation in Context

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- The Meaning of Certain Words in Math
- Single Step Equations
- Multi-Step Equations
- Solving Problems in Context

**The Meaning of Certain Words in Math**

Word problems can be tricky. Often it takes a bit of practice to convert the English sentence into a mathematical sentence. This is what we will focus on here with some basic number problems, geometry problems, and parts problems. A few important phrases are describes below that can give us clues for how to set up a problem.

*Word to Math Examples*

- A number (or unknown, a value, etc.) often becomes our variable
- Is (or other forms of is: was, will be, are, etc.) often represents equals (=). For example "x is give" becomes "x = 5"
- "More than" often represents addition and is usually built backwards, writing the second part plus the first. For example, "three more than a number" becomes "x + 3"
- "Less than" often represents subtraction and is usually built backwards as well, writing the second part minus the first. For example, "four less than a number" becomes "x – 4"
- The product of two numbers often represents multiplication between two quantities. For example "the product of two unknowns is 5" means that we have x • y = 5
- The quotient of two numbers often represents division between two quantities. For example, "the quotient of a number and 5 is 4" can be written as 5 / x = 4

**Single Step Equations**

Suppose we were told that Rachel earned $40,000 a year for some number of years and ended up making a total of $280,000 over that time period. For how many years did Rachel work? Can you determine an equation to represent the problem and solve the equation?

Here, we know how much Rachel earns each year, but we are told that this was for "some number of years," which alerts us that the number of years worked would be a variable. We can also think that if we multiply the number of years worked with her salary each year, we can set that number equal to the total amount earned. Using this equation, we can then solve the problem.

**Multi Step Equations**

Using these key phrases, we can take a number problem and set up an equation and solve:

This same idea can be extended to a more involved problem as shown in the next example:

**Solving Problems in Context**

When working with word problems, it is always a good idea to start by clearly labeling the variables in a short list before we begin to solve the problem. This is important in all word problems involving variables, not just consecutive numbers or geometry problems. This is shown in the following example:

Be careful on problems such as these. Many students see the phrase "double" and believe that it means we only have to divide 444 by 2 and get S = 222 for one or both of the prices. As you can see, this will not work. By clearly labeling the variables in the original list, we know exactly how to set up and solve these problems.

Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html