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A variable is a symbol used to represent a number.
The letters x and y are the most common variables used by textbooks. Technically, any symbol can be used (even a smiley face if you like). Variables simplify calculations and make work easier to read.
An algebraic expression consists of at least one variable and at least one arithmetic operation. An arithmetic operation is either addition, subtraction, multiplication, or division.
2x - 1 is an algebraic expression because we have at least one variable and one arithmetic operation (subtraction).
The following are all examples of algebraic expressions:
Note: x + 1 means "add the value 1 to some number x". We don't know what number x is, but we know that whatever x is, x + 1 is one more than x.
Consider the algebraic expression x + 1. Suppose we let x = 1. Note: x = 1 means that we can replace x with the value 1. So x + 1 is equal to 1 + 1 (just throw out the x and replace it with the value 1), but we know that 1 + 1 is equal to 2. We write our results like this: Let x = 1. Then x + 1 = 1 + 1 = 2. If we let x = 3, then we have that x + 1 = 3 + 1 = 4.
Look at some of the different ways to write x multiplied by y:
| xy | x y |
(x)y | x(y) | (x)(y) |
In each expression above, the quantities being multiplied are called factors and the result is called the product.
Often we need to translate verbal expressions into algebraic expressions when dealing with real life
problems. For example, translate the following verbal expression into an algebraic expression:
"two times a number x plus two"
Solution: "two times a number x" translates
as 2x, then we have plus 2 which we can write as + 2. So the answer is 2x + 2.
There are several different ways to express the four arithmetic operations with words. We will
list three for each arithmetic operation, but just know that there are many more. For
example, a + b can be expressed in words as "The sum of a and b", "a plus b", or "add b to a".
The algebraic expression ab can be expressed in words as "a times b", "the product of a and b",
or "multiply a and b". The algebraic expression a - b can be expressed as "the difference of
b from a", "subtract b from a", or "a minus b". The algebraic expression
can be expressed as "the quotient of a and b", "a divided by b",
or "a over b".
An expression like 2n is read as "2 raised to the power of n". This means that
2 is going to be multiplied by itself n times. For example, suppose we let n = 2.
Then 2n = 22 = 22 = 4. If we let n = 3,
then we have that 2n = 23 =
222 = 8.
33 = 33
3
3 12 = 11
| Algebraic Expression | Verbal Expression | Meaning |
| x1 | x raised to the power of one or x raised to the first power | x |
| x2 | x raised to the power of two or x raised to the second power or x squared | xx |
| x3 | x raised to the power of three or x raised to the third power or x cubed | xxx |
| 2x4 | 2 times x raised to the power of four or 2 times x raised to the fourth power | 2xxxx |
1. 2x + 2, x = 0
2. 4 - 2x, x = 1
3. x2, x = 3
4. 3x + 4, x = 1
5. Three subtracted from x.
6. Two times x plus 3.
7. The product of x and y subtracted from one.
8. Two times the product of x and y.
9. Three times the sum of 3 and b.
10. The product of 5 and x plus the product of 3 and y.
11. Two times x squared subtracted from 3.
12. The sum of 3 and a divided by b.
13. John grew two inches. (Let j = John's height before he grew)
14. John lost two pounds. (Let j = John's weight before losing 2 pounds)
15. x - 4
16. x2 + y2
17. 3m3 - 2
18. (1/2)n + x
19. x2 + 1
20. xx + 2
21. xyyy
22. 101010
23. 23
24. 140
25. 72
26. 43
27. Use your calculator to evaluate the expressions in problems #23-26, and see if
your answer matches what your calculator says.