Algebra Notes

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# Algebra Notes - PowerPoint PPT Presentation

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.

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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

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

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

Solve

Check

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

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

Evaluate if s = 4

• 4s →
• 4 + s →
• 5 – s →
• 12 ÷ s →

4(4)

16

4 + 4

8

5 - 4

1

12 ÷ 4

3

Evaluate if s = -6

• 7s →
• 3 + s →
• 7 – s →
• 18 ÷ s →

7(-6)

-42

3 + (-6)

-3

7 – (-6)

13

18 ÷ (-6)

-3

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

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

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

Terms of an Expression

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

3x + 5y – 8 has 3 terms.

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

LIKE terms: Yes or No?

3x + 7x

Yes - Like

5x + 5y

No - Unlike

4c + c

Yes - Like

4d + 4

No - Unlike

LIKE terms: Yes or No?

3ab – 6b

No - Unlike

2a – 5a

Yes - Like

x andx²

No - Unlike

Yes - Like

6 and 10

Identify the LIKE terms

3m – 2m + 8 – 3m + 6

5x + b – 3x + 4 + 2x – 1 – 3b

-6y + 4yz + 6x² + 2yz – 4y + 2x² - 5

Coefficients

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

Example: 6x

The coefficient is 6.

Example: x

The coefficient is 1.

Simplify

• Simplify: means to combine like terms.
• Combine LIKE terms by adding their coefficients.

Write an expression:

-

5a – 4b

This expression cannot be simplified. Why not?

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

1. –5x – 3x 2. 8x – 2x

3. –7x – (–3x) 4. 6x – (–4x)

5. –10x –14x 6. –9x – (–x)

7. 3x – 8x 8. x – (–5x)

9. a² + b² + 2a² + 5b² 10. 7h² + 3 – 2h² + 4

-8x

6x

-4x

10x

-8x

-24x

-5x

6x

5h² + 7

3a + 6b²

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

• 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: _________+ __________+ _________

2x² - 3x²

6x – 2x

5 – 1

5x²

4x

4

5x² + 4x + 4

Subtracting

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

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: _________+ __________+ _________

-

-

+

+

5y² - 2y²

2xy – 3xy

-5 + 3

3y²

-xy

-2

3y² - xy - 2

• 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

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

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
• 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

Practice #1

3(m - 4)

3 • m - 3 • 4

3m – 12

Practice #2

-2(y + 3)

-2 • y + (-2) • 3

-2y + (-6)

-2y - 6

Simplify the following:

3(x + 6) =

3x + 18

4(4 – y) =

16 – 4y

7(2 + z) =

14 + 7z

5(2a + 3) =

10a + 15

Simplify the following:

6(3y - 5) =

18y – 30

3 +4(x + 6) =

4x + 27

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

17x – 4

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

12cd

4x

2a

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)

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 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 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
• 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

x – 3 = 19 2. a – 14 = 6

• 3. 9x = 63 4. 5x – 2 = 8
• 6. 8a + 5 = 53
• -7 = c – 6 8. a – 3.5 = 4.9
• 9. x – 2.8 = 9.5 10. 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

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

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 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 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 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 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 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 & 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

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?
• 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:
• 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.
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?

Does not equal…

Example: x ≠ 7 means:

7 is not equal to x

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 < 3
• x > -5
• x < -1
• x > 2
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, & Check
• 5n + 4 < 4n
• 3x – 3 ≤ 9

x < -6

y ≥ 3

n < -4

x ≤ 4