Find practice problems and solutions for these concepts at Explaining Variables and TermsPractice Problems.
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.
Algebra Vocabulary
A sentence, whether it contains variables or not, is made up of terms. A term is a variable, constant, or product of both, with or without exponents, and is usually separated from another term by addition, subtraction, or an equal sign. While a variable can have different values in different situations, a constant is a term that never changes value. Real numbers are constants. The sentence x + 4 = 5 contains 3 terms: x, 4, and 5.x is a variable, and 4 and 5 are constants.
The sentence 3x – 5 = 11 also contains 3 terms: 3x –5, and 11.3x is a single term, because 3 and x are multiplied, not added or subtracted. In the same way, – = 2 also has only three terms, because is a single term.
The numerical multiplier, or factor, of a term is called a coefficient. In the term 3x, 3 is the coefficient of x, because 3 multiplies x. In the term 9y, the coefficient of y is 9. In multiplication, the order in which one value multiplies another does not matter: 4×5and 5×4 both equal 20. The order in which 3 and x are multiplied does not matter, either, but we typically place the constant in front of the variable. The constant is considered the coefficient, and the variable is considered the base. Because the coefficient is one factor of the term, the base is the other factor. If a variable appears to have no coefficient, then it has a coefficient of 1: 1x = x.
Tip:
Division can be rewritten as multiplication. x divided by 5 is the same as x multiplied by The coefficient of x in the term is because can be written as
In algebra, the base of a term is often raised to an exponent. An exponent is a constant or variable that tells you how many times a base must be multiplied by itself. Exponents are small numbers (superscripts) that appear above and to the right of a base. The term x2 is equal to x multiplied by x. The term y6 means (y)(y)(y)(y)(y)(y). If a variable appears to have no exponent, then it has an exponent of 1: x1 = x.
Like and Unlike Terms
If two terms have the same base raised to the same exponent, then the two terms are called like terms. For instance, 2a2 and –6a2 are like terms, because both have a base of a with an exponent of 2. Even though the terms have different coefficients, they are still like terms. If two terms have different bases, or identical bases raised to different exponents, then the two terms are unlike terms. 7m and 7n are unlike because they have different bases. 7m4 and 7m5 are also unlike terms. Even though they have the same base, the exponents of the bases are different. In the next lesson, we will see why recognizing terms as like or unlike is so important.
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