Algebra notes
This presentation is the property of its rightful owner.
Sponsored Links
1 / 86

Algebra Notes PowerPoint PPT Presentation


  • 89 Views
  • Uploaded on
  • Presentation posted in: General

Algebra Notes. Writing Algebraic Expressions. Let Statement: math sentence used to define a variable to represent the unknown quantities. Laura has twice as much homework as Ann. The Bills won five more games than they lost. Seven more than three times a number is 25.

Download Presentation

Algebra Notes

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Algebra notes

Algebra Notes


Writing algebraic expressions

Writing Algebraic Expressions


Algebra notes

Let Statement:math sentence used to define a variable to represent the unknown quantities.


Algebra notes

Laura has twice as much homework as Ann.

The Bills won five more games than they lost.

Seven more than three times a number is 25.

The length of a rectangle is 3 cm more than the width.

Let Ann = a

Let Laura = 2a

Let games lost = g

Let Bills won = 5 + g

Let Yankees = y

Let Tigers = 3y

Let width = w

Let length = 3 + w


Algebra notes

Mike is three years older than Jim.

Eight more than twice a number is 32

Seven more than three times a number is 25.

Twice a number increased by four is 16.

Let Jim = j

Let Mike = 3+ j

Let number = n

2n + 8 = 32

Let number = n

3n + 7

Let number = n

2n + 4 = 16


Algebra notes

Six less than three times a number is 21.

Fifteen less than twice a number is 25.

Sixty-six is eleven more than five times a number.

Let number = n

3n – 6 = 21

Let number = n

2n – 15 = 25

Let number = n

66 = 5n + 66


Writing algebraic expressions1

Writing Algebraic Expressions


Setting up solving word problems

Setting up & Solving word problems


Algebra notes

Write your let statement

Write your equation

Solve

Check

Write an answer sentence


Algebra notes

A cell phone company charges $39 a month plus $.15 per text message sent. If Jan sends 35 text messages this month, how much does she owe before taxes are added?

The Bills won five more games than they lost.

Let text message = t

Jam owes $44.25

39 + 0.15t

t = 44.25

Let text message = s

12 + 2s

s = 4

4 snacks


Algebra notes

A rental car company ABC charges $25 per day plus $.15 per mile. Rental car company XYZ charges $18 per day plus $.25 per mile. If you plan to drive 50 miles, who is the cheaper rental company?

Joe attends a carnival. The admission is $48. Tickets for rides cost $4 each. Joe needs one ticket for each ride. Write an equation Joe can use to determine the number of ride tickets, r, he can buy if he has $200 before he pays the admission fee.

Let miles = m

ABC: 25 + 0.15m

$32.50

XYZ: 18 + 0.25

$30.50

XYZ is cheaper

Let number or rides = r

48 + 4r = 200

r = 38

38 rides


Substitutions

substitutions


Algebra notes

Evaluate if s = 4

  • 4s →

  • 4 + s →

  • 5 – s →

  • 12 ÷ s →

4(4)

16

4 + 4

8

5 - 4

1

12 ÷ 4

3


Algebra notes

Evaluate if s = -6

  • 7s →

  • 3 + s →

  • 7 – s →

  • 18 ÷ s →

7(-6)

-42

3 + (-6)

-3

7 – (-6)

13

18 ÷ (-6)

-3


Algebra notes

Evaluate if n = 3 and r = 5

  • n² + 7r

  • 9n - r²

  • 2nr + 6n

3² + 7(5)

9 +12

21

9(3) - 5²

27 – 25

2

2(3)(5) + 6(3)

30 + 18

48


Algebra notes

Evaluate if p = 12 and q = -8

  • p + q +6

  • p – q + 3

  • p – q + q²

12 + (-8) + 6

-4 + 6

2

12 – (-8) + 3

20 + 3

23

12 – (-8) + (-8)²

20 + 64

84


Algebra notes

Evaluate if a = -2 and b = 6

  • 3a² + 5b²

  • 4a³ + 3b

  • 7a² - (b²/3)

3(-2)² + 5(6)²

3(4) + 5(36)

192

4(-2)² + 3(6)

4(-8) + 18

-14

7(-2)² - (6²/3)

7(4) – (36/3)

28 – 12

16


Like terms

Like Terms


Algebra notes

Terms of an Expression

  • Termsare parts of a math expression separated by addition or subtraction signs.

3x + 5y – 8 has 3 terms.


Algebra notes

Like Terms

  • Like Terms: have thesame variablesto the same powers

8x²+2x²+5a +a

8x²and 2x² are like terms

5a and a are like terms


Algebra notes

LIKE terms: Yes or No?

3x + 7x

Yes - Like

5x + 5y

No - Unlike

4c + c

Yes - Like

4d + 4

No - Unlike


Algebra notes

LIKE terms: Yes or No?

3ab – 6b

No - Unlike

2a – 5a

Yes - Like

x andx²

No - Unlike

Yes - Like

6 and 10


Algebra notes

Identify the LIKE terms

3m – 2m + 8 – 3m + 6

5x + b – 3x + 4 + 2x – 1 – 3b

-6y + 4yz + 6x² + 2yz – 4y + 2x² - 5


Algebra notes

Coefficients

  • A Coefficient: a numberwritten in front of the variable.

Example: 6x

The coefficient is 6.

Example: x

The coefficient is 1.


Algebra notes

Simplify

  • Simplify: means to combine like terms.

  • Combine LIKE terms by adding their coefficients.


Algebra notes

Write an expression:

+

3c + 4c

=7c


Algebra notes

Write an expression:

-

8a - 1a

= 7a


Algebra notes

Write an expression:

+

5c + 4d


Algebra notes

Write an expression:

-

5a – 4b

This expression cannot be simplified. Why not?


Simplify the following expressions

Simplify the following expressions


Algebra notes

  • 2x + 4x

  • 2a + 5a + 6

  • 3xy – xy +2x

  • -4c + 8c – 6c

  • 3a + 7a

  • 3½y + 5y -4y

  • cd + 4cd – 2a

  • ½e – 2e + ¾ e

  • 6xy – 2xy

  • 5d – 6d – 3d

  • 4s – 4s

  • 5x + 4x + 4x + 11x

6x

10a

4xy

4½y

7a + 6

-4d

2xy + 2x

5cd – 2a

0

-¾e

24x

-2c


Challenge questions

Challenge questions


Algebra notes

1.–5x – 3x2.8x – 2x

3.–7x – (–3x)4.6x – (–4x)

5.–10x –14x6.–9x – (–x)

7.3x – 8x8.x – (–5x)

9. a² + b² + 2a² + 5b²10. 7h² + 3 – 2h² + 4

-8x

6x

-4x

10x

-8x

-24x

-5x

6x

5h² + 7

3a + 6b²


Algebra notes

11. 3x + 3y + x + y + z 12. 5b +5b + 6b² - 10 – 3b

13. Find the perimeter of the rectangle:

A 4x + 3y

B 8x + 6y

C 12xy

D 4x²+ 3y²

4x + 4y +z

6a² + 7b - 10


Adding subtracting polynomials

Adding & subtracting Polynomials


Adding

Adding

  • Combine like term

  • Add the coefficients to simplify

    Example: Add   2x² + 6x + 5   and  3x² - 2x – 1

    • Start with:2x² + 6x + 5     +3x² - 2x – 1

    • Place like terms together:_______+ ________+ ________

    • Add the like terms: _________+ __________+ _________

    • Final answer:

2x² - 3x²

6x – 2x

5 – 1

5x²

4x

4

5x² + 4x + 4


Subtracting

Subtracting

Change the subtraction sign to addition and reverse the sign of each term that follows

Then add as usual

Example: Subtract   5y² + 2xy - 5   and  3x² - 2x – 1

Start with:5y² + 2xy - 5     -2y² - 3xy + 3

Place like terms together:_______+ ________+ ________

Add the like terms: _________+ __________+ _________

Final answer:

-

-

+

+

5y² - 2y²

2xy – 3xy

-5 + 3

3y²

-xy

-2

3y² - xy - 2


Try the following

Try the following


Algebra notes

  • 1. (2x + 3y) + (4x + 9y) 2.(3a + 5b + 7c) - (5a – 2b + 9c)

  • 3. (3x – 5) + (x – 7) + (7x + 12)

  • 4.(3a + 5b + 7c) + (8a – 2b – 9c)

  • 5. –4x³ + 6x² – 8x – 10 and 7x³ – 4x² + 9x + 3

  • 6. Subtract (5m – 6n + 12) from (2m + 3n – 5).

  • (2m + 3n – 5) - (5m – 6n + 12)

  • -2a + 7b – 2c

    6x + 12y

    11x

    11a + 3b – 2c

    3x³ + 2x² - x - 7

    -3m + 9n -17


    Algebra notes

    7.Subtract 8a + 5b – 6c from 10a + 8b + 7c

    (10a + 8b + 7c) - (8a + 5b – 6c)

    8. (4x + 8y + 9z – 7a + 5b) – (4b + 5x + 7y + 3z + 2a)

    9. (– 3x2 + 4x – 11) – (–6x2 – 8x + 10).

    10. (7e² + 3e +2) + (9 – 6e + 4e²) + (9e + 2 – 6e²)

    2a – 3b + 13c

    -x – y + 6z – 9a + b

    3x² + 12x - 21

    5e² + 6e + 13


    Challenge

    challenge


    Algebra notes

    Some of the measures of the polygons are given. P represents the measure of the perimeter. Find the measure of the other side or sides.

    x² - 15x + 3

    2x + y

    4x - 3

    14x² - 4x + 7


    The distributive property

    The distributive property


    The distributive property1

    The Distributive Property

    • Distributive Property: the process of distributing the number on the outside of the parentheses to each term in the inside.

    a(b + c) = ab + ac

    Example:

    5(x + 7) =

    5x + 35

    5•x

    +

    5•7


    Algebra notes

    Practice #1

    3(m - 4)

    3 • m - 3 • 4

    3m – 12

    Practice #2

    -2(y + 3)

    -2 • y + (-2) • 3

    -2y + (-6)

    -2y - 6


    Algebra notes

    Simplify the following:

    3(x + 6) =

    3x + 18

    4(4 – y) =

    16 – 4y

    7(2 + z) =

    14 + 7z

    5(2a + 3) =

    10a + 15


    Algebra notes

    Simplify the following:

    6(3y - 5) =

    18y – 30

    3 +4(x + 6) =

    4x + 27

    2x + 3(5x - 3) + 5 =

    17x – 4


    Distributive practice

    Distributive practice


    Algebra notes

    • 2(4 + 9x) 2. 7(x + -1) 3. 12(a + b + c)

    • 7(a + c + b) 5. -10(3 + 2 + 7x) 6. -1(3w + 3x + -2z)

    • -1(x + 2) 8. 3(-2 + 2x2y3 + 3y2) 9. 5(5 + 5x)

    • y(1 + x) 11. 12x(3x + 3) 12. 9(9x + 9y)

    8 + 18x

    7x - 7

    12a + 12b + 12c

    -3w – 3x + 2z

    7a + 7c + 7b

    -70x - 50

    -x – 2

    25x + 25

    -6 + 6x²y³ + 9y²

    y + yx

    36x² + 36x

    81x + 8y


    Factoring

    Factoring


    Factoring1

    factoring

    • To factor expressions find the GCF (greatest common factor) of the terms

    • Factoring is the opposite of distributing.


    Find the gcf of each pair of monomials

    Find the GCF of each pair of monomials

    • 4x, 12x 2. 18a, 20ab 3. 12cd, 36cd

    12cd

    4x

    2a


    Factor each expression

    Factor each expression

    4. 12a – 6h 5. 3x + 9 6. 12x + y

    7. 24a – 4 8. 72a + 9n 9. 8a - 8v

    3(x + 3)

    6(2a – h)

    Cannot be simplified

    4(6a – 1)

    9(8a + n)

    8(a – v)


    Solving equations

    Solving equations


    Steps to solving equations

    Steps to Solving Equations

    • Equation: a mathematical sentence that uses an equal (=) sign.

    • Step 1: Get rid of the 10. Look at the sign in front of the 10, since it is subtraction we need to use the opposite operation (addition) to cancel out the 10

      • Add 10 to both sides. Remember, what you do to one side of the equation, you have to do to the other.

    2n – 10 = 50

    +10

    +10

    2n = 60


    Steps to solving equations1

    Steps to Solving Equations

    • Step 2: Next, we need to look at what else is happening to the variable. 2n means that two is being multiplied to n, therefore we need to do the opposite (division) to “undo” the multiplication.

      • Divide both sides by 2. Remember, what you do to one side of the equation, you have to do to the other.

    2n = 60

    2

    2

    n

    = 30


    Steps to solving equations2

    Steps to Solving Equations

    • Step 3: CHECK your solution!! First, rewrite the original equation

      • We already solved for n, so wherever you see the variable, n, plug in the answer.

      • Evaluate the equation, SHOWING ALL WORK!

      • Does it check?

    2n – 10 = 50

    2 (30) – 10 = 50

    60 – 10 = 50

    50 = 50


    Solve check

    Solve & Check

    • 105 = 10n + 5

    • n/5 + 3 = 6

    • -44 + 7n = 250

    • -1/2 = -5/18h

    • 200 = 100 – 25n

    • -9.4 + z = -3.6

    n = 10

    n = 15

    n = 42

    h = -9/5

    n = -4

    z = 5.8


    Solving equations practice

    Solving equations practice


    Algebra notes

    • x – 3 = 192.a – 14 = 6

    • 3. 9x = 634.5x – 2 = 8

    • 6. 8a + 5 = 53

    • -7 = c – 68.a – 3.5 = 4.9

    • 9. x – 2.8 = 9.510. 2.25 + b = 1

    • .

    x = 22

    a = 22

    x = 22

    x = 22

    x = -30

    a = 22

    c = 22

    a = 22

    x = 12.3

    b = 22

    14.2(b – 2) + b + 3


    Algebra notes

    • 11.12. -8.5 + r = -2.1

    • 13. 14. 2(b – 2) + b = 6.5

    • .

    r = 6.4

    c = 1 3/7

    m = 33/14

    b = 2.5


    Solving multi step equations

    Solving multi-step equations


    Steps to solving multi step equations

    Steps to Solving Multi-Step Equations

    • Step 1: Distribute if necessary variable.

      • Distribute the 4 to the n and 5.

    4(n – 5) - 7 = 9 + 2n – 4n

    4n – 20 - 7 = 9 + 2n – 4n


    Steps to solving multi step equations1

    Steps to Solving Multi-Step Equations

    • Step 2: Combine like terms on each side of the equations.

      • On the left side -20 and -7 combine to get -27

      • On the right side 2n and -4n combine to get -2n

    4n – 20 - 7 = 9 + 2n – 4n

    4n – 27 = 9 – 2n


    Steps to solving multi step equations2

    Steps to Solving Multi-Step Equations

    • Step 3: Get all variables to one side of the equation.

      • First we want to get rid of the -27. Look at the sign in front of -27, since it is subtraction (or a negative) we need to use the opposite operation (addition) to cancel it out. Therefore add 27 to both sides.

    4n – 27 = 9 – 2n

    +27

    +27

    4n = 36 – 2n


    Steps to solving multi step equations3

    Steps to Solving multi-step Equations

    • Step 4: Get all “plain numbers” to one side of the equation

      • First we want to get rid of the -2n. Look at the sign in front of -2n, since it is subtraction (or a negative) we need to use the opposite operation (addition) to cancel it out. Therefore add 2n to both sides.

    4n = 36 – 2n

    +2n

    +2n

    6n = 36


    Steps to solving multi step equations4

    Steps to Solving Multi-Step Equations

    • Step 5: Next, since we have all the variables on one side and all the “plain numbers” on the other side we need to look at what else is happening to the variable.

      • 6n means the 6 is being multiplied by n, therefore we need to do the opposite (division) to “undo” the multiplication. So, divide both sides by 6.

    6n = 36

    6

    6

    n = 6


    Steps to solving multi step equations5

    Steps to Solving Multi-Step Equations

    • Step 6: CHECK your solution!! First, rewrite the original equation

      • We already solved for n, so wherever you see the variable, n, plug in the answer.

      • Evaluate the equation, SHOWING ALL WORK!

      • Does it check?

    4(n – 5) - 7 = 9 + 2n – 4n

    4(6 – 5) - 7 = 9 + 2(6) – 4(6)

    4(1) - 7 = 9 + 12 – 24

    4 – 7 = 21 - 24

    -3 = -3


    Solve check1

    Solve & Check

    • 9 + 5r = -17 – 8r

    • 3(n + 5) + 2 = 26

    • 58 + 3y = -4y – 19

    • 4 – 2(v – 6) = -8

    r = -2

    n = 3

    y = -11

    v = 12


    Inequalities

    Inequalities


    Algebra notes

    Inequality: a mathematical sentence using <, >, ≥ , or ≤.

    Example: 3 + y> 8.

    Inequalities use symbols like <and> which means less than or greater than.

    They also use the symbols ≤ and ≥which means less than or equal to and greater than or equal to.

    Inequalities


    What s the difference

    What’s the difference?

    • x < 4 means that x is less than 4

      • 4 is not part of the solution

      • What number is in this solution set?

    • x ≤ 4 means that x can be less than OR equal to 4

      • 4 IS part of the answer

      • What number is in this solution set?


    You graph your inequalities on a number line

    You graph your inequalities on a number line:

    • This graph shows the inequality x < 4

    • The open circle on 4 means that’s where the graph starts, but 4 is NOT part of the graph.

    • The shaded line and arrow represent all the numbers less than 4.


    What is this inequality

    What is this inequality?

    X > -2


    What is this inequality1

    What is this inequality?

    X ≥ 2 1/2


    Graphing inequality solution sets on a number line

    Graphing inequality solution sets on a number line:

    • Use an open circle ( ) to graph inequalities with < or > signs.

    • Use a closed circle( ) to graph inequalities with ≥ or ≤ signs.


    What do you think this symbol means

    What do you think this symbol means?

    Does not equal…

    Example: x ≠ 7 means:

    7 is not equal to x


    Graph x 1

    Graph x ≠ -1

    • X ≠ -1 would include everything on the number line EXCEPT -1.

    • Use an open circle to show that -1 is NOT a part of the graph.


    Graph x 4 a number less than 4

    Graph x < 4 (a number less than 4)


    Graph x 6 a number less or equal to 6

    Graph x < 6 (a number less or equal to 6)


    Graph x 3 a number less or equal to 6

    Graph x >3 (a number less or equal to 6)


    Graph each inequality

    Graph each inequality


    Graph

    Graph

    • x < 3

    • x > -5

    • x < -1

    • x > 2


    Solve graph and check each inequality

    Solve, Graph, and check each inequality


    Solve graph check

    Solve, Graph, & Check

    • x + 8 > 15

    • 3y – 4 < 11

    • 2x < 18

    • x + 4 > 2

    • 2n + 7 > 13

    x > 7

    y < 5

    x < 9

    x > -2

    n > 3


    Solve graph check1

    Solve, Graph, & Check

    • 5n + 4 < 4n

    • 3x – 3 ≤ 9

    x < -6

    y ≥ 3

    n < -4

    x ≤ 4


  • Login