1. Objectives Chapter 6 Writing Algebra Powers Square Roots
The Number Line
Adding Signed Numbers
Subtracting Signed Numbers
Multiplying Signed Numbers
Dividing Signed Numbers
Evaluating Expressions
Writing Expressions
Using Formulas
Understanding Equations
One-step Equations
Two-step Equations
Equations with Separated Unknowns
Equations with Parentheses
Using Formulas Like Equations

2. Algebra
Unit 6
Pages 114 – 142

3. Writing Algebra
Algebra is arithmetic written with letters as well as numbers.
Recall:
In arithmetic, the times sign (x) means to multiply.
In algebra, multiplication is shown using a raised dot or by writing the two numbers or symbols next to each other.
There are four basic number operations: addition, subtraction, multiplication, and division.
The answer to an addition problem is called the sum. The plus sign (+) shows addition.
The answer to a subtraction problem is called the difference. The minus sign (-) means to subtract.
Perhaps the most important symbol in arithmetic or algebra is the equal sign (=). In mathematical statements, it often stands for the word is.

4. Example Write each of the following with numbers and symbols.
The sum of eight and seven
The product of nine and five

5. Group Work Write each of the following with numbers and symbols.
The sum of one-half and three
The quotient of ten divided by two
The sum of twelve and three is fifteen

6. Powers Algebra uses an operation called raising a number to a power.
82 means “eight to the second power.” The number 8 is called the base. The small number 2 is called the exponent.
To find a power:
Write the base the number of times the exponent tells you.
Multiply them.
Raising a number to the second power is also called squaring a number.
82 is eight to the second power or eight squared.

7. Example
Solve
102 = =

8. Group Work
Solve
42 =
82 – 22 =
103 – 52 =
82 – 52 =
42 – =

9. Square Roots
The opposite of powers is called roots. The opposite of finding a second power is finding a square root. For example, 5 to the second power is 25. The square root of 25 is 5. The symbol for square root is

10. Example
Solve =
36 − 4 =

11. Group Work
Solve
100 − 25 =
16 +52= =
144 −12=

12. The Number Line
Algebra sometimes uses numbers that are less than zero. These are called negative numbers. The number line shows both positive and negative numbers.
All the whole numbers, decimals, and fractions that are greater than zero are positive numbers.

13. Group Work Plot the numbers on the number line. 3 –5 1 ¼ 6 –3 ½ –1
7 3/4

14. Adding Signed Numbers
To add numbers with the same signs, add and give the answer the sign of the numbers.
To add numbers with different signs, subtract. Then give the answer the sign of the number with the greater absolute value.

15. Example
Solve.
–6 + 9 =
(–20) + (–19) =

16. Group Work Solve. –8 + (+9) = (–13) + (–12) = –18 + 21 = +3 + (–5) =
– =

17. Subtracting Signed Numbers
To subtract signed numbers, change the sign of the number being subtracted and the subtraction sign to an addition sign.
Then add the signed numbers.

18. Example
Solve.
–6 – (–10) =
5 – (+12) =

19. Group Work Solve. –13 – (–9) = (–25) – (30) = –21 – (–8) =
32 – (+12) =

20. Multiplying Signed Numbers
When you multiply two signed numbers:
The answer is positive if the signs are alike.
The answer is negative if the signs are different.

21. Example
Solve.
–9(–3) =
–12(7) =

22. Group Work
Solve.
–4(–8) =
(+5)(–11) =
3(+16) =
–10(–9) =
36(–3) =

23. Dividing Signed Numbers
When you divide two signed numbers:
The answer is positive if the signs are alike.
The answer is negative if the signs are different.

24. Example
Solve.
35/–7 =
–18/–3 =

25. Group Work
Solve.
–27/9 =
–12/–24 =
–250/–25 =
–15/35 =
–72/–8 =

26. Evaluating Expressions
An expression is a mathematical operation or instruction written with numbers and symbols. To evaluate means to find the value.
Mathematicians have agreed on the following order for performing operations when an expression calls for more than one operation.
Order of Operations:
Parentheses and division bars
Powers and roots
Multiplication and division from left to right
Addition and subtraction from left to right

27. Example
Solve.
50/2 + 3(10) =
c – d when c = 12 and d = 5

28. Group Work Solve. 7 + x when x = 3 2(x + 4) when x = 8
3a + 4b when a = 6 and b =2
5a when a = –7
4a2 when a = 5

29. Writing Expressions
Algebra is a tool used for expressing number relationships.

30. Example Write an expression for each of the following.
Three times a number
Thomas makes p dollars a month. He tries to save one-tenth of his income. Write an expression for the amount he saves each month.

31. Group Work Write an expression for each of the following.
Nine less than a number
Helen is y years old. Her daughter Katie is 23 years younger. Write an expression for Katie’s age.
Eddie makes w dollars a week. He works 35 hours every week. Write an expression for the amount e makes each hour.
Gloria’s gross salary is g dollars. Her employer deducts 21% of her gross salary for taxes. Write an expression for the amount of the deductions.

32. Using Formulas
A formula is a mathematical instruction written in the language of algebra.
The distance formula d = rt means “distance equals rate times time,” where
d is distance, usually measured in miles
r is rate, usually measured in miles per hour (mph) and
t is time, usually measured in hours.

33. Example Use the distance formula to solve these problems.
Kareem drove for 2.5 hours at an average speed of 60 mph. How far did he drive?
Deborah jogs at an average of 6.5 mph. How far can she job in two hours?

34. Group Work Use the distance formula to solve these problems.
A plane flew at an average speed of 418 mph. How far did it travel in 3.5 hours?
Michelle is driving across the country. On the first day of her trip, she plans to drive six hours. If she averages 65 miles per hour, how far will she drive the first day?
Driving a truck through a mountainous area, Stan estimates he can drive at an average rate of 30 miles per hour. If he drives for 4 ½ hours at that rate, how many miles will he travel?

35. Understanding Equations
An equation is a statement that two amounts are equal.

36. Example
For each equation written in words, choose the corresponding equation written in numbers and symbols.
The sum of twelve and x is nineteen
a) 12x b) 12 + x c) 12 + x = 19 d) 12x = 19
The number n decreased by seven is equal to fifteen.
a) n – 7 = 15 b) n/7 = 15 c) 15n = 7 d) 7n = 15

37. Group Work
For each word equation, write an equation in numbers and symbols.
The number s decreased by thirteen is equal to twenty-one.
Nine is equal to two times z.
The sum of p and six is equal to fifty.
The product of x and twelve is equal to one hundred eight.
Eight times y is equal to fifty.

38. One-Step Equations
The solution to an equations is the value of the unknown that makes the statement true.
To solve equations, use opposite or inverse operations. Do these operations on both sides of the equation.
Subtraction is the inverse of addition
Addition is the inverse of subtraction
Division is the inverse of multiplication
Multiplication is the inverse of division

39. Example
Solve each equation.
41 = b – 12
d/13 = 2

40. Group Work Solve each equation. 9a = 72 16 = h/5
Sixteen is equal to a number divided by four. Find the number.
Tiffany makes $22 dollars an hour for over-time work. This is twice her regular wage. Find her regular wage.
Carlos saved $28 buying a sweater on sale. The sale price was $35. Find the original price of the sweater.

41. Two-Step Equations
To solve two-step equations, use inverse operations in the following order:
Do addition or subtraction operations first.
Do multiplication or division operations last.

42. Example
Solve each problem.
8m + 7 = 55
5a – 3 = 17

43. Group Work Solve each problem. x/6 + 3 = 7 37 = 4c + 9
Five less than twice a number is nine. Find the number.
Ten more than three time a number is twenty- two. What is the number?
Four less than one-half of a number is seven. Find the number.

44. Equations with Separated Unknowns
When the unknowns of an equation are separated, combine them according to the rules of addition of signed numbers.
Like terms have the same variables

45. Example
Solve each problem.
5a – a = 24
6y – 7 – 5y = 14

46. Group Work Solve each problem. 8e = 30 + 5e 3h = 12 + 2h
8n – 9 = 7n + 13
Ten times the number n equals three times the same number n increased by seven. Find the number
Seven times a number n increased by twelve equals twice the number n decreased by eight. Find the number.

47. Equations with Parentheses
Before you can solve an equation with parentheses, you need to multiply to get rid of the parentheses.

48. Example
Solve each problem.
3(a + 2) = 18
9(c – 2) = c + 30

49. Group Work Solve each problem. 6(y – 5) = 42 3(r + 4) = 2r + 17
Six times the difference of n and two equals eighteen. Find the value of n.
Seven times the sum of n and five equals fifty- six. Find the value of n.

50. Using Formulas Like Equations
You have used the distance formula d = rt to calculate distance. You can also use the formula to calculate the rate when you know both distance and time.
If you know any two variables, you can solve the distance formula like an equation to find the third variable.

51. Example Use the distance formula d=rt to solve the following.
Keith drove 160 miles in 2.5 hours. What was his average driving speed?
On the highway, Phyllis sticks to the speed limit of 65 miles per hour. How long will it take her to reach a destination 260 miles away?

52. Group Work
The formula c = nr is the formula for finding total cost when you know the number (n) you are buying and the rate (r), the cost for one.
Frank paid $7.20 for 2.5 pounds of beef. What was the price per pound (the rate)?
Mira paid $38.85 for three gallons of paint. What was the cost of one gallon?
Sliced ham was priced at $2.90 a pound. Emma paid $4.35 for the amount of ham she needed. Find the weight of the ham she bought.
Marleen paid $19.20 for 12 feet of lumber. What was the price per foot?