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# Explaining Variables and Terms Study Guide

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## Introduction to Explaining Variables and Terms

The human mind has never invented a labor-saving machine equal to algebra.

—Author Unknown

In this section, you'll learn the language of algebra, how to define variables and terms, and a short review of integers.

Math topics always seem to have scary sounding names: trigonometry, combinatorics, calculus, Euclidean plane geometry —and algebra. What is algebra? Algebra is the representation of quantities and relationships using symbols. More simply, algebra uses letters to hold the place of numbers. That does not sound so bad. Why do we use these letters? Why not just use numbers? Because in some situations, we do not always have all the numbers we need.

Let's say you have 2 apples and you buy 3 more. You now have 5 apples, and we can show that addition by writing the sentence 2 + 3 = 5. All of the values in the sentence are numbers, so it is easy to see how you went from 2 apples to 5 apples.

Now, let's say you have a beaker filled with 134 milliliters of water. After pouring more water into the beaker, you look closely and see that you now have 212 milliliters of water. How much water was added to the beaker? Before you perform any mathematical operation, that quantity of water is unknown.

#### Tip:

If we do not know the value of a quantity in a problem, that value is an unknown.

We can write a sentence to show what happened to the volume of water n the beaker even though we don't know how much water was added. A symbol can hold the place of the quantity of water that was added. Although we could use any symbol to represent this quantity, we usually use letters, and the most commonly used letter in algebra is x.

There is no clear reason why x came to be used most often to represent unknowns. René Descartes, a French mathematician, was one of the first to use x, y, and z to represent unknown quantities—back in 1637! While many have tried to determine why he used these letters, no one knows for certain.

The beaker had 134 milliliters of water in it when x milliliters were added to it. Read that sentence again. We describe the unknown quantity in the same way we would a real number. When a symbol, such as x, takes the place of a number, it is called a variable. We can perform the same operations on variables that we perform on real numbers. After x milliliters are added to the beaker, the beaker contains 212 milliliters. We can write this addition sentence as 134 + x = 212. Later in this book, we will learn how to solve for the value of x and other variables.

In the sentence 134 + x = 212, 134 and 212 are numbers and x is a variable. Because the variable x holds the place of a number, we can perform the same operations on it that we would perform on a number.

We can add 4 to the variable x by writing x + 4. We can subtract 4 from x by writing x – 4. Multiplication we show a little bit differently. Because the letter x looks like the multiplication symbol (×), we show multiplication by placing the number that multiplies the variable right next to the variable, with no space. To show 4 multiplied by x, we write 4x. There is no operation symbol between 4 and x, and that tells us to multiply 4 and x. Multiplication is sometimes shown by two adjacent sets of parentheses. Another way to show 4 multiplied by x is (4)(x).

Division is most often written as a fraction. x divided by 4 is . This could also be written as or x ÷4, but these notations are less common.

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