unit 2: equations€¦ · unit 2: equations name: ___ teacher: _ period: ___ 2 swbat: solve...

1

Unit 2: Equations

Name: ___________

Teacher: __________

Period: _____

2

SWBAT: Solve one step equations and one step literal equations.

DO NOW: Solve.

1) x + 29 = 59 2) n – 12 = 25 3) -8x = -96 4) 12

x_ = 3

When Solving Equations the Goal is to Get the Variable by Itself.

I) Simplify each side of the equation. Remember ____ separates it into 2 sides.

(Same side use _____________ operation)

II) Get variable terms on one side and _______________ on the _______________ side.

(Opposite side use _______________ (inverse) operations.)

III) Solve for the variable

3-Step Check: 1) Rewrite 2) Replace 3) PROVE

Literal Equation -- an equation in which known quantities are expressed with letters

Formula -- A type of literal equation that shows a relationship between quantities • When you use a literal equation you may be asked to solve it for one variable in terms of

the others

• To do this--pretend that all the other variables except for the one you are solving for

are numbers and proceed as follow the steps for solving equations.

Example 1:

Solve & Check Literal Equation

8.5 + y = 10.2 Solve for y: x + y = 12

3

Example 2:

Solve Literal Equation

n - 8 = -12 Solve for a: a – 7 = b

Example 3:


-6x = 24 Solve for t: d = rt

Example 4:


8

x = -5 Solve for a: q

p

a =

4

Math 8: HW Solving one step equations and one step literal equations.

Solve each equation algebraically and 3-step check each one.

1) x – 13 = -48 2) x + 26 = -13

3) x-8 = -24 4) -7x = 56

Solve each of the following for the indicated variable.

5) Solve for w: w + r = 5 - 3r 6) Solve for w: V = l w h

7) Solve for t: I = prt 8) Solve for x: x + c = 2b

9) Solve for x: 3a = 2c + x 10) Solve for x: 4c + x = a

11) Solve for x: bx = 2cd

5

AIM: SWBAT solve two-step equations and two step literal equations.

“Do Now” Solve each of the following for the indicated variable. 1) Solve for l: A = lw 2) Solve for d: C = πd

3) Solve for x: y

x = a 4) Solve for x: y = x + b

SOLVING EQUATIONS

GOAL: Get the __________________ by itself.

What separates one side of the equation from the other side? _____________ I) ______________ each side of the equation. (No parenthesis, no like terms) II) Get the variable term on one side and the

constant term on the ______________ side.

(Opposite side use ______________ operation) III) Solve for the variable by using the ______________ (opposite) operation

Example 1:


5d + 17 = -68 Solve for x: 2x + y = 6

Example 2:


-15y – 16 = 59 Solve for y: -3y – x = b

6

Example 3:


2-

n+ 6 = 12 Solve for h:

2

h + d = A

Example 4:


2

6 n += 12 Solve for x:

4

x a += L

Solve each of the following for the indicated variable:

1) Solve for w: P = 2l + 2w 2) Solve for b: A = 2

bh

7

Math 8: HW Solving two step equations and two step literal equations.

Solve each equation algebraically and check #1 and #2.

1) 2x - 3 = 1 2) -x + 4 = 1 3) -3x + 6 = 12 4) -9 = -2y – 1 5) 7y - 4 = -11 6) -9 = 7 – x

7) 78a2

3 =− 8) 1268x −=−

9) 35

7x −=− 10)

53+a

= -11

8

Solve each of the following for the indicated variable. 11) Solve for x: 3a = 2c - x 12) Solve for l: P = 2l + 2w

13) Solve for m: y = mx + b 14) Solve for x: azy

x =−

15) Solve for x: Tg

m-x = 16) Solve for y: 2x + 3y = 7

9

AIM: SWBAT translate phrases into mathematical expressions.

“DO NOW”

Write the symbol of the operation that pertains to each word or group of words. 1) decreased by _______________ 2) product ______________

3) increased by _____________ 4) less than _____________

5) quotient _________________ 6) sum _______________

7) more than ________________ 8) difference ________________

9) of _______________ 10) twice _______________

Translate each phrase into a mathematical expression.

** 11) five more than twelve __________________________

** 12) six less than twenty ________________________

13) the product of seven and y ______________

14) fifteen divided by five _______________

15) twenty decreased by fifteen _______________

16) nine increased by one _______________

17) the difference of four and two _______________

18) eight less than x _______________

19) the sum of five and y _______________

20) twelve more than nine _______________

21) the quotient of twelve and four _______________

22) twice nine _______________

23) the product of three and y _______________

24) the sum of nine and eight _______________

25) the quotient of x and 12 _______________

**26) twice a number, n, increased by x _______________

**27) twice the sum of x and y _______________

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Aim: SWBAT define a variable, write an equation, and solve word problems.

Remember you define a variable by identifying what it is you are looking for in the problem.

For Example: Find the number. Let x = the number How much does one shirt cost? Let x = cost of one shirt Define a variable, write an equation, and solve each word problem.

Your final answer must be a complete sentence.

1) Thirty-six more than a number is 47. 2) Forty-seven less than a number is 21. Find the number. Find the number. 3) Keilah has 58 points. Seventy-two points 4) A total of 4,635 runners started the are needed to win the game. How many marathon. Only 2,338 runners finished. more points does she need to win? How many runners did not finish the race? 5) Four times a number plus 75 is 23. 6) Six less than twice a number is -14. Find the number. Find the number. 7) Frank bought two shirts for $38. He 8) Colleen spent $105 on two pairs of shoes. paid $4 more for one than for the One pair cost $5 more than the other other. How much did he pay for each pair. How much did she pay for each shirt? pair of shoes?

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Homework: Define a variable, write an equation, and solve each word problem. Final answer must be a complete sentence. 1) A coat and pants cost $78.50. If the 2) If twice a number is subtracted from 8, cost of the coat was $31.70 more than the difference is 12. the cost of the pants, how much did What is the number? each item cost? 3) Laura bowled a 197 game last week. 4) The total cost of 8 tickets to a play This is 14 pins more than her average. was $360. What was the cost of What is her average? each ticket?

5) Three times a number plus -8 is -29. 6) Angela scored 15 points less than twice Find the number. the lowest score on a math test. If her score was 95, what was the lowest score on the test?

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AIM: SWBAT solve multi-step equations.

Solving equations our GOAL is to get the ________________ by itself.

What separates one side of the equation from the other side?_______________

Get rid of any parentheses. How do we get rid of parentheses? ___________________

You need to simplify each side of the equation by ________________________

We need all VARIABLE TERMS on one side & _____________ TERMS on the OPPOSITE

SIDE.

We move terms to OPPOSITE sides using ______________________ operations.

Solve each equation algebraically. Check #1 and 2.

1) 5(3x – 2) + 11 = 106 CHECK 2) -2x + 6x – 9 = 15 CHECK

3) 2(3x + 2) + 8 = 6 4) 7y – 3(2y + 1) = 12

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5) -4(x + 5) = 35 + 25 6) -7(x - 3) = -7 Let’s try some multi-step literal equations. 7) Solve for r: a(l + r) = b 8) Solve for k: 3(k – 5) = g

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Math 8: HW Solving multi-step equations and multi-step literal equations.


1) x + 3(x – 4) = 16 CHECK: 2) -8(x - 8) = -40 CHECK:

3) 5(3 – 2n) + 5n = 65 4) 5n + 3(n – 4) = 28

5) 7y – 3y – 8 = -32 6) -22 + 3k + 6 = -28

Solve each of the following for the indicated variable.

7) Solve for b: A = 2(b + d) 8) Solve for x: r = 5(x + 2y)

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Aim: More Practice with One- and Two-Step

Solve each of the following. Be sure to show all steps!

1) 3m – 2 = 16 2) 2m + 5 = -29 3) 4 + 8y = 20

4) 4y – 8 = 0 5) 743

y=+ 6) 31x

32 =−

7) 21

41

b =+− 8) 6 – m = -4 9) 45

2x=

+

10) 53

b6−=

+ 11) 3y + 4 = 11 12) 11 – y = 10

16

Solve each of the following for the indicated variable. Show your steps!

13) I = 12F for F 14) 4c – x = a for c 15) g

mxT

−= for x

16) I = prt for t 17) W + L = 5 – 3L for W 18) 2x + 3y = 7 for y

19) d = rt for r 20) X + 4c = 2B for X 21) X + 4c = 2B for C

22) X + 4c = 2B for B 23) F = 5C + 30 for C 24) 2x + 3y = 7 for x

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AIM: SWBAT solve equations with variables on both sides.

DO NOW: Fill in the answers to #1 - 5

1) Our goal is to get the _____________________ by itself.

2) What separates one side of the equation from the other side? ____________

3) You need to ______________ each side of the equation by getting rid of parenthesis and

combining like terms.

4) We need all VARIABLE TERMS on one side and all ______________TERMS on the

OPPOSITE side.

5) We move terms to the OPPOSITE sides using ______________ operations.

Solve each equation algebraically. Check #1 and 2.

1) 16x = 4x + 36 2) 2x + 16 = 8x – 26

3) 14x – 19 = 5x + 8 4) 6(x + 2) = 4x + 22

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5) 2(x – 4) = 4x – 2 6) 12x + 8 – 4x = 2(x + 16) Let’s try some literal equations with variables on both sides. 7) Solve for x: ax + b = 3b 8) Solve for x: 3x – q = 5q 9) Solve for y: cy – d = 4d 10) Solve for x: 9x - 24a = 6a + 4x

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Math 8: HW Solving multi-step equations with variables on both sides and multi-step literal

equations.


REMEMBER: We need variable terms on one side and constant terms on the opposite side!

1) 5x = 8 + 3x CHECK: 2) 24x = 27x – 6 CHECK: 3) 8 + 6x = 2x 4) 0.4x - 36 = 0.5x 5) x + 28 = 13x – 8 6) 10x - 2 = 8x - 1

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7) 2(3x-10) = 40 8) 3(x-5) = 2(2x+1) 9) 6x - 8 = 2(2x + 1) 10) 4(x+5) - 5(2x-1) = x-24 Solve each of the following for the indicated variable.

11) Solve for b: b + 5r = 7r 12) Solve for a: ab – c = 3c

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SWBAT: Solve word problems with variables on both sides.

“Do Now”

Solve & Check

7y – 5 = 9y + 29

Define a variable, write an equation, and solve each word problem.


1) If 7 times a number is decreased by 33, the result is 15 less than 10 times the number. Find the number.

2) The greater of two numbers is 6 times the lesser. The greater number is 25 more than

the lesser. Find the numbers.

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3) If a number is doubled and then increased by 20, the result is 5 less than 3 times the number. Find the number.

4) The greater of two numbers is three times the lesser. If the greater number is decreased by 12, the result is 6 more than the lesser. Find the numbers.

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Math 8 - Homework Word Problems using equations with variables on both sides Define a variable, write an equation, and solve each word problem.


1) When two times a number is increased by 10, the result is 1 more than 3 times the

number. Find the number.

2) The greater of two numbers is 7 more than the lesser. Three times the greater number is 5 more than 4 times the lesser number. Find the numbers.

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3) The greater of two numbers is twice the lesser. If the greater is increased by 18, the result is 4 less than 4 times the lesser. Find the numbers.

4) The greater of two numbers is 1 less than 4 times the lesser. Three times the lesser number is 4 less than the greater. Find the numbers.

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SWBAT : identify and solve identity and contradictions equations.

When solving an equation, we are looking for the value of the variable that will make the

equation true statement.

Ex: 2x + 4 = 10

-4 = -4

2x = 10

2 2

x = 5

Sometimes, a linear equation can also have no solution, or infinite number of solutions.

One Solution No Solution Infinite Number of Solutions

2x = x + 1

-x -x x = 1 Only one number makes the equation true.

x + 1 = x + 2

-x -x

1 2

Since 1 2 can never be true, there are NO numbers that can make the equation true. This type of equation is called a CONTRADICTION.

x + 3 = x + 3

-x -x 3 = 3 Since 3 = 3 is always true, you can substitute ANY number for x and the equation will be true. This type of equation is called an IDENTITY. x is any Real number.

Determine if the following equations have a solution, no solution (Contradiction) or infinitely many solutions (Identity):

1) 10x - 15 = 5(2x + 4) 2) 3(2x+ 4) = 6x + 12 3) -4(x + 2) = 12 – 2x

3) 3x + 7 = 3(x + 2) 4) 2x + 9 = 2(x + 4) + 1

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Determine whether the following equations are an identity or a contradiction equation:

5) c + d = d + c _______________ 6) 71

+ y = y _______________

7) 4x + 6 = 4(x + 1)_______________ 8) 21

x = 42

x _______________

9) 6 = 9 _______________ 10) 0 = 0 _______________ 11) 0 = -5 _______________ 12) x + 7 = x - 7 _______________ 13) 4a – 6 = 6 + 4a _______________ 14) 2(x + 7) = 2x + 14 _______________ 15) a + b = -b + a _______________ 16) 2x – 3 = 2x - 3 _______________ 17) Create an equation that is a CONTRADICTION. ______________________________ 18) Create an equation that is an IDENTITY. ______________________________

Homework

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Identities and Contradictions

An Identity—An equation that is true for all values of the unknown. It has an infinite number of

solutions. X is any Real number.

A Contradiction—An equation having no solution.

Solve each equation and CHECK your solution, if possible. If the equation is an Identity or a

Contradiction you must explain what that means in terms of the solution.

1) 3(y + 4) = y + 1 + 2y 2) 34x – 5(2x + 1) = 16x + 19

3) 6x – 2x + 4 = 4(x + 1) 4) 4(y – 9) = 3( y + 4) 5) 6x + 9 – x = 5(x – 4)

6) 2x – 4(x + 2) = -2x – 8 7) 4(x + 5) – 5(2x – 1) = x - 24

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AIM: SWBAT solve multi – step equations involving fractions.

Solve each equation algebraically.

1) 9

8x = 2 2)

74

x + 5 = -19

3) 95

x + 7 = 91

x - 5 4) 37

x + 6 = 35

x

29

5) 3

2x -

5

3 =

10

7 6)

6

1(12x + 24) =

3

1(15x + 21)

7) 2x + 8

1 =

41

(4x - 1) 8) 4

1 -

3

2x =

4

3 -

3

1x

30

Math 8: HW solve multi – step equations involving fractions.



1) -53

+ x = 201

2) 21

(4x + 12) = x - 2

3) 52

(10x + 35) = 81

(16x + 80) 4) 75

x - 12 = -47

31

5) 5

1x - 1

3

1 = -2

3

2 -

5

3x 6)

4

3x - 7 =

2

1x + 3

7) 32

x + 41

x = 22 8) 31

x + 2 = -x + 4

13

32

Aim: SWBAT solve multi-step fraction equations by “clearing out” the fractions.


1) 9

8x = 2 2)

74

x + 5 = -19

3) 95

x + 7 = 91

x - 5 4) 37

x + 6 = 35

x

5) 32

x - 53

= 10

7 6)

41

- 32

x = 43

- 31

x

33

Be careful with "clearing out” fractions AND using the distributive property.

Do NOT Double Distribute!!

7) 41

(4x - 1) = 2x + 8

1

8) 61

(12x + 24) = 31

(15x + 21)

34

Math 8: HW solve each multi – step equation by “clearing out” the fractions.

REMEMBER: Do NOT double Distribute

1) -5

3 + x =

20

1 2)

4

3x - 7 =

2

1x + 3

3) 75

x - 12 = -47 4) 51

x - 131

= -232

-53

x

35

REMEMBER: Do NOT double Distribute

5) 32

x + 41

x = 22 6) 31

x + 2 = -x + 4

13

7) 2

1(4x + 12) = x - 2 8)

5

2(10x + 35) =

8

1 (16x + 80)

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AIM: SWBAT solve multi – step equations involving decimals.


1) 10-

x - 2.4 = -2.2 2) -3x – 1.5 = 1.35

3) 1.2x + 6.7 = 0.139 4) 0.28 - y = -6.9

37

5) 3.5x + 6 = 1.5x 6) 0.02(x + 5) = 8 7) 0.4(x - 9) = 0.3(x + 4) 8) 0.2(2x - 3) = 0.16x - .36

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Math 8: HW solve multi – step equations involving decimals.



1) 3.8 - 0.5x = 0.3 2) 0.25x + 3.4 = 1.6

3) 0.8x + 6 = 0.5x - 9 4) 0.03y – 1.2 = 8.7

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5) 1.2(2x + 4) = 7.92 6) 0.6x – 0.5 = 6.124 7) 0.05x + 10 = 0.06(x + 50) 8) 0.8x + 0.35 = 7.55

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Aim: SWBAT solve multi-step decimal equations by “clearing out” the decimal.

Do Now

Solve each of the following equations algebraically. If it is a “special” equation, identify

what kind it is, state how many solutions there are and what the solution is.

1) 31

(12x - 9) = 4x – 3 2) 337x34 =− 3) 2(x + 6) = 2x + 9

Solve each equation by “clearing out” the decimals.

REMEMBER: Do NOT Double Distribute.

1) 10-

x - 2.4 = -2.2 2) -3x – 1.5 = 1.35

3) 1.2x + 6.7 = 0.139 4) 0.28 - y = -6.9

41

5) 3.5x + 6 = 1.5x 6) 0.02(x + 5) = 8 7) 0.4(x - 9) = 0.3(x + 4) 8) 0.2(2x - 3) = 0.16x - .36

42

Math 8: HW solve multi – step equations by “clearing out” the decimals.

REMEMBER: Do NOT Double Distribute.

1) 3.8 - 0.5x = 0.3 2) 0.25x + 3.4 = 1.6

3) 0.8x + 6 = 0.5x - 9 4) 0.03y – 1.2 = 8.7

43

5) 1.2(2x + 4) = 7.92 6) 0.6x – 0.5 = 6.124 7) 0.05x + 10 = 0.06(x + 50) 8) 0.8x + 0.35 = 7.55

44

Review solving equations. Solve each of the following ALGEBRAICALLY. Show all work.

1) 2

w183

−=− 2) Solve for x: 5x + 6 = 2x + y 3) 1n

3

18n

3

2 −=+

4) 3( y + 4) = y + 1 + 2y 5) 0.71 -0.22x = 1.03 – 0.62x 6) 15x + 5 = 5(3x + 1)

7) -5 – x = 3x + 5 8) Solve for a: c2

ba=

+ 9) a + 11 = 21 – a

10) 9x – 3 = 4x + 12 11) 3x + 1 = 4x – 11 12) 1x41

5x43 −=−

45

13) -5( 1 - x) = 3x + 2 14) 2.5q + 8 = 6.5q – 4 15) 3(y + 1) = 4y – 1 16) 6x – 2x + 15 = 4 (x + 1) 17) 11(x – 2) = 3(x + 6) 18) 2x – 4(x + 2) = -2x -8 19) Solve for y: 3x + 2y = 6 20) In the equation 4 (x – 6) = -44 + 3x does x = -20? 21) In the equation 3 (2x – 5x) = x + 45 does x = -4.5? 22) Becky is five years older than Brian. The sum of their ages is 87. Find their ages.

46

Since d=rt:

d=(55)(6)

d=330 miles

SWBAT: Use formulas to solve problems.

A formula is an equation which states a rule that shows the relationship between certain quantities.

Example #1

The formula Distance=Rate x Time can be used to calculate how far, how fast, or how long something travels.

If: d = distance r = rate t = time ⇒ We can use the formula d=rt.

If Joe is driving on the Thruway at 55 miles per hour for 6 hours, how far has he traveled?

We know that: rate = 55mph time = 6 hours distance = ? He has traveled 330 mile.

Example #2

A person’s typing speed in words per minute can be calculated by using the following formula:

n

10ews

−= Where: s = speed in words per minute w = number of words typed

e = number of errors made n = number of minutes typed

If Angela typed 285 words in 5 minutes and made 6 errors, how many words per minute does she type?

Complete the following problems using the formulas given: 1) The formula used to calculate the maximum heart rate during a workout is:

5

4a)(220B

•−= B= beats per minute and a= age in years

a) Calculate the maximum heart rate in beats per minute for a 15 year old. b) Calculate the maximum heart rate in beats per minute for a 40 year old.

47

2) A formula used to calculate the fare for a taxi cab ride is: F = .75 + 1.5d F = taxi fare and d = distance in miles

If you had to take a taxi to school one morning and your house is 4.5 miles from school, what would your cab fare be? 3) The earned run average of a pitcher in baseball is calculated by dividing the earned runs by the number of innings pitched. Write a formula you could use to calculate the earned run average using the following variables: ERA = earned run average i = number of innings pitched e = earned runs 4) One side of a square field is 140 meters long. What is the perimeter of the field?

5) Using the formula F = 5

9C + 32, where F= degrees Fahrenheit and C= degree Celsius;

a) If it is 25 degrees Celsius, what is the corresponding temperature in degrees Fahrenheit? b) If it is 30 degrees Fahrenheit, what is the corresponding temperature in degrees Celsius?

48

HW: Using formulas to solve problems.

1) Find the temperature on the Fahrenheit scale that corresponds to 20° Celsius.

Use: F = 5

9C + 32

2) Find the temperature on the Celsius scale that corresponds to 50° Fahrenheit.

Use: F = 59C + 32

3) Find the length of a rectangle if the perimeter is 68in. and the width is 14in. Use: P = 2l + 2w

4) Find the width of a rectangular prism if the Volume is 106m3, the length is 3m and the height is 2m.

Use: V = lwh

5) Find the height of a triangle if the Area is 32ft2 and the base is 4ft.

Use: A = 2

bh

49

REVIEW 2 – MATH 8R – EQUATIONS

Circle: TRUE or FALSE.

1) –x + 5 = 12, if x = -7 2) -20 = -7x - 1, if x = 3 3) -8(x - 12) = 32, if x = 8

TRUE FALSE TRUE FALSE TRUE FALSE

State the first step to solve each equation.

4) Colleen solved the equation -8

1x - 10 = -8 and says the answer is x = 16. Do a formal

check and decide if her answer is correct or incorrect. Colleen’ s answer is: (circle one) CORRECT or INCORRECT

Write the following statements as expressions only

5) Twelve more than a number “y” 6) Nine more than twice a number “x”

7) Five less than three times a number “g” 8) Six times the sum of y and 24.

Solve the following equations algebraically.

9) 43

x = 36 10) 94

x + 19 = -37 11) 32 (

54

x + 15) = 51 (

31

x – 55)

12) 8

5x +

6

5 =

24

10− 13) -1.2(x – 8) = 14.4 14)

5

2x7 +− = 20


15) Solve for w: v = 3

hwl •• 16) Solve for x:

4

yx −= w 17) Solve for r: rs - 9 = x

50

Solve each equation if possible. State whether the equation is an Identity or a Contradiction and explain what that means in terms of the solution.

18. 3

1 (12x – 24) = 4x - 8 19. 7(x + 2) = 5(x + 3) + 2x

Solve each of the following equations by clearing the fractions or decimals:

20. 41

(-16x + 32) = 21

(4x – 32) 21. -0.3(x – 5) = -9.9

Define a variable(s), write an equation and solve each word problem. Final answer is a sentence.

22) When one-half a number is increased by 9, the result is -15. Find the number. 23) A sweater and jeans cost a total of $97. The sweater cost $7 more than the jeans. Find the cost of each item. 24) The larger of two numbers is 12 more than the smaller. If the larger is increased by 20, the result is 8 less than 2 times the smaller. Find the numbers.

25) Shoe size and foot length are related by the formula: FS =+

3

24, where

S=shoe size F = foot length

Find the length of a person’s foot if their shoes are a size 6.

unit 2: equations€¦ · unit 2: equations name: _____ teacher: _____ period: _____ 2 swbat: solve...

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unit 2: equations€¦ · unit 2: equations name: ___ teacher: _ period: ___ 2 swbat: solve...