1-1 Variables and Expressions

Quick Review

A variable is a symbol, usually a letter, that represents values of a variable quantity. For example, d often represents distance. An algebraic expression is a mathematical phrase that includes one or more variables.

A numerical expression is a mathematical phrase involving numbers and operation symbols, but no variables.

Example

What is an algebraic expression for the word phrase 3 less than half a number x?

You can represent “half a number x” as x 2 . x over 2 , .  Then subtract 3 to get x 2 − 3 . x over 2 , minus 3 .

Exercises

Write an algebraic expression for each word phrase.

6. the product of a number w and 737
7. the difference of a number q and 8
8. the sum of a number x and 84
9. 9 more than the product of 51 and a number t
10. 14 less than the quotient of 63 and a number h
11. a number b less the quotient of a number k and 5

Write a word phrase for each algebraic expression.

12. 12 + a
13. r − 31 r minus 31
14. 19t
15. b ÷ 3 b divides 3
16. 7 c − 3 7 c minus 3
17. 2 + x 8 2 plus , x over 8
18. y 11 − 6 y over 11 , minus 6
19. 21d + 13

1-2 Order of Operations and Evaluating Expressions

Quick Review

To evaluate an algebraic expression, first substitute a given number for each variable. Then simplify the numerical expression using the order of operations.

1. Do operation(s) inside grouping symbols.
2. Simplify powers.
3. Multiply and divide from left to right.
4. Add and subtract from left to right.

Example

A student studies with a tutor for 1 hour each week and studies alone for h hours each week. What is an expression for the total hours spent studying each week? Evaluate the expression for h = 5.

The expression is h + 1. To evaluate the expression for h = 5, substitute 5 for h: (5) + 1 = 6.

Exercises

Simplify each expression.

20. 9 2 9 squared
21. 5 3 5 cubed
22. ( 1 6 ) 2 open , 1 sixth , close squared
23. 7 2 ÷ 5 7 squared , divides 5
24. ( 2 4 − 6 ) 2 open , 2 to the fourth , minus 6 close squared
25. ( 3 3 − 4 ) + 5 2 open , 3 cubed , minus 4 close plus , 5 squared

Evaluate each expression for c = 3 and d = 5.

26. d 3 ÷ 15 d cubed , divides 15
27. ( 2 + d ) 2 − 3 2 open 2 plus d close squared . minus , 3 squared
28. c d 2 + 4 c , d squared , plus 4
29. ( 3 c 2 − 3 d ) 2 − 21 open 3 , c squared , minus 3 d close squared . minus 21
30. The expression 6 s 2 6 , s squared  represents the surface area of a cube with edges of length s.
    1. What is the cube's surface area when s = 6?
    2. Reasoning Explain how a cube's surface area changes if you divide s by 2 in the expression 6 s 2 . 6 , s squared , .
31. A race car travels at 205 mi/h. How far does the car travel in 3 h?

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