Lesson 1 Algebraic Properties of Equality and Identity

NCSCOS Obj.: 1.01, 1.02

ObjectiveTLW recognize and use the properties of identity and

equality.

Identity Properties

1) Additive Identity

What do you add to get the same?

a + 0 = a

2) Multiplicative Identity

What do you mult. to get the same?

a • 1 = a

Inverse Properties

1) Additive Inverse (Opposite)

a + (-a) = 0

2) Multiplicative Inverse (Reciprocal)

a 1

a 1

Multiplicative Property of Zero

a • 0 = 0

(If you multiply by 0, the answer is 0.)

Properties of Equality1) Reflexive: a = a

5 = 5

2) Symmetric: If a = b then b = a.

If 4 = 2 + 2 then 2 + 2 = 4.

3) Transitive:If a = b and b = c, then a = c.

If 4 = 2 + 2 and 2 + 2 = 3 + 1 then 4 = 3 + 1.

4) Substitution: If a = b, then a can be replaced by b.

(5 + 2)x = 7x

Name the Property1. 0 12 = 0

Multiplicative Prop. Of Zero2. (10 + 2) 3 = 12 3

Substitution3. 2 + 3 = 5 then 5 = 2 + 3

Symmetric

4. If 5 2 = 10 & 10 = 5 + 5 then 5 2 = 5 + 5

Transitive

5. 6 + (-6) = 0

Additive Inverse

6. 1 m = mMultiplicative Identity

7. k + 7 = k + 7 Reflexive

8. x + 0 = x Additive Identity

9.

111

11 1

Multiplicative Inverse

Name the property.0 + 4 = 4

1. Additive Identity

2. Additive Inverse

3. Additive Property of Zero

4. Substitution

Answer NowAnswer Now

Name the property.8 – (6 + 2) = 8 - 8

1. Additive Identity

2. Additive Inverse

3. Associative

4. Substitution

Answer NowAnswer Now

Name the property.2 + (x – 3)1 = 2 + (x – 3)

1. Reflexive

2. Multiplicative Inverse

3. Multiplicative Identity

4. Symmetric

Answer NowAnswer Now

Commutative PropertyCommutative means that the order

does not make any difference.a + b = b + a a • b = b • a

Examples

4 + 5 = 5 + 4

2 • 3 = 3 • 2

The commutative property does not work for subtraction or division.

Associative PropertyAssociative means that the grouping

does not make any difference.(a + b) + c = a + (b + c) (ab) c = a (bc)

Examples

(1 + 2) + 3 = 1 + (2 + 3)

(2 • 3) • 4 = 2 • (3 • 4)

The associative property does not work for subtraction or division.

Name the property

1) 5a + (6 + 2a) = 5a + (2a + 6)

commutative (switching order)

2) 5a + (2a + 6) = (5a + 2a) + 6

associative (switching groups)

3) 2(3 + a) = 6 + 2a

distributive

Which property would justify rewriting the following expression without parentheses?

3(2x + 5y)1. Associative property of

multiplication2. Distributive property3. Addition property of

zero4. Commutative property

of multiplication

Answer Now

Which property would justify the following statement?

8x + 4 = 4 + 8x1. Associative property of

addition

2. Distributive property

3. Addition property of zero

4. Commutative property of addition

Answer Now

Which property would justify the following statement?

8 + (2 + 6) = (8 + 2) + 6 1. Associative property of

addition

2. Distributive property

3. Addition property of zero

4. Commutative property of addition

Answer Now

The Distributive PropertyThe process of distributing the number

on the outside of the parentheses to each term on the inside.a(b + c) = ab + ac and (b + c) a = ba + ca

a(b - c) = ab - ac and (b - c) a = ba - ca

Example #1

5(x + 7)

5 • x + 5 • 7

5x + 35

Example #2

3(m - 4)3 • m - 3 • 4

3m - 12

Example #3

-2(y + 3)-2 • y + (-2) • 3

-2y + (-6)-2y - 6

Which statement demonstrates the distributive property incorrectly?

1. 3(x + y + z) = 3x + 3y + 3z

2. (a + b) c = ac + bc

3. 5(2 + 3x) = 10 + 3x

4. 6(3k - 4) = 18k - 24

Answer Now

A term is a1) number,

2) variable, or

3) a product / quotient of numbers and variables.

Example

5

m

2x2

The coefficient isthe numerical part of the term.

Examples1) 4a

4

2) y2

1

3) 5x2

7 5

7

Like Terms are terms with the same variable AND exponent.

To simplify expressions with like terms, simply combine the like terms.

Are these like terms?

1) 13k, 22k

Yes, the variables are the same.

2) 5ab, 4ba

Yes, the order of the variables doesn’t matter.

3) x3y, xy3

No, the exponents are on different variables.

Which of the following is the simplified form of

-4x + 7x ?1. -4

2. 3x

3. -3x

4. 4

Answer Now

8x 2 2x2 5a a

The above expression simplifies to:

10x2 6a

8x 2 2x2

5a and a are like terms

and are like terms

12a

2) 6.1y - 3.2y

2.9y

3) 4x2y + x2y

5x2y

4) 3m2n + 10mn2 + 7m2n - 4mn2

10m2n + 6mn2

Simplify1) 5a + 7a

21a + 6b

6) 4d + 6a2 - d + 12a2

18a2 + 3d3y

4

y

47)

3y

4

1y

4

4y

41y

y

5) 13a + 8a + 6b

Which of the following is the simplified form of

5x - 4 - 7x + 14 ?

Answer Now

1. -12x + 10

2. -2x + 10

3. 2x - 18

4. 12x – 18

If a triangle has sides 3x - 2, 5 - x and 2x - 5, what is the perimeter of the triangle?

Answer Now

1. 4x - 2

2. 4x + 2

3. 5x - 2

4. 5x - 7

Which figure below models the simplification of - 4x - 5 + 7x + 7 using these tiles?

Answer Now

1.

2.

3.

4.

Bonus! Which of the following is the simplified form of a + 3a - 4(9 - a) ?

Answer Now

1. -36

2. 3a - 36

3. 8a - 36

4. 8a + 36