Lesson 2.1 Solving Equations w/Justification

Concept: Solving Equations EQ: How do we justify how we solve equations? REI. 1 Vocabulary: Properties of Equality Properties of Operation Justify 1

Solve the equations below, provide an explanation for your steps.

1.

2x – 3 = 13 2.

3𝑥+1 = 5 2 2

Properties of Equality

Property In symbols

Reflexive property of equality Symmetric property of equality Transitive property of equality

a

=

a

If

a

=

b

, then

b

=

a.

If

a

=

b

and

b

=

c

, then

a

=

c

.

Addition property of equality If

a

=

b

, then

a

+

c

=

b

+

c

.

Example

2=2 x = 3 3 = x x = 2, y = 2, x = y

x – 4 = 3 x – 4 + 4 = 3 + 4 x = 7

2.1.1: Properties of Equality

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Properties of Equality, continued

Property In symbols

Subtraction property of equality If

a

=

b

, then

a

–

c

=

b

–

c

.

Multiplication property of equality If

a

=

b

and

c

≠ 0, then

a

•

c

=

b

•

c

.

Division property of equality If

a

=

b

and

c

≠ 0, then

a

÷

c

=

b

÷

c

.

Examples

x + 2 =5 x + 2 – 2 = 5 – 2 x = 3 x=15

4x = 16 x = 4

2.1.1: Properties of Equality

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Properties of Equality, continued

Property In symbols

Substitution property of equality If

a

=

b

, then

b

may be substituted for

a

in any expression containing

a

.

Examples x = 3, then 2x = 2(3) = 6

2.1.1: Properties of Equality

5

Properties of Operations

Property General rule Specific example

Commutative property of addition

a

+

b

=

b

+

a

Associative property of addition (

a

+

b

) +

c

=

a

+ (

b

+

c

3 + 8 = 8 + 3 ) (3 + 8) + 2 = 3 + (8 + 2) Commutative property of multiplication

a

•

b

=

b

•

a

3 • 8 = 8 • 3 Associative property of multiplication Distributive property of multiplication over addition (

a a

• • (

b b

) • +

c c

= ) =

a a

• ( •

b b

+ •

a c

) •

c

(3 • 8) • 2 = 3 • (8 • 2) 3 • (8 + 2) = 3 • 8 + 3 • 2

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2.1.1: Properties of Equality

Guided Practice Example 1

Which property of equality is missing in the steps to solve the equation –7

x

+ 22 = 50?

Equation

–7

x

+ 22 = 50 –7

x

= 28

x

= –4

Steps

Original equation Division property of equality 2.1.1: Properties of Equality

7

Guided Practice: Example 1, continued 1.

Observe the differences between the original equation and the next equation in the sequence. What has changed?

Notice that 22 has been taken away from both expressions, –7x + 22 and 50.

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2.1.1: Properties of Equality

Guided Practice: Example 1, continued 2.

Refer to the table of Properties of Equality.

The subtraction property of equality tells us that when we subtract a number from both sides of the equation, the expressions remain equal.

The missing step is “Subtraction property of equality.” ✔

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2.1.1: Properties of Equality

Guided Practice: Example 1, continued

2.1.1: Properties of Equality

10

Guided Practice Example 2

Which property of equality is missing in the steps to solve the equation −3 − 𝑥 6 = 4?

Equation Steps

Original equation −𝑥 = 7 6 –

x

= 42

x

= –42 Addition property of equality Division property of equality 2.1.1: Properties of Equality

11

Guided Practice: Example 2, continued 1.

Observe the differences between the original equation and the next equation in the sequence. What has changed?

Notice that 3 has been added to both expressions, −3 − 𝑥 6 and 4. The result of this step is − 𝑥 6 = 7 .

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2.1.1: Properties of Equality

Guided Practice: Example 2, continued

In order to move to the next step, the division of 6 has been undone.

The

inverse operation

of the division of 6 is the multiplication of 6.

x

The result of multiplying by 6 is –x and the result 6 of multiplying 7 by 6 is 42. This matches the next step in the sequence.

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2.1.1: Properties of Equality

Guided Practice: Example 2, continued 2.

Refer to the table of Properties of Equality.

The multiplication property of equality tells us that when we multiply both sides of the equation by a number, the expressions remain equal.

The missing step is “Multiplication property of equality.” ✔

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2.1.1: Properties of Equality

Guided Practice: Example 2, continued

2.1.1: Properties of Equality

15

Guided Practice: Example 3

What equation is missing based on the steps?

1.

Observe the 3 rd and 5 th equations.

2.

Read the 4 th step.

3.

Fill in the missing equation. 2.1.1: Properties of Equality

16

You Try… Identify the property of equality that justifies each missing step or equation.

Equation

9 + x = 17 x = 8 3.

Steps

Original Equation

Equation

7(2x + 1) = 49 14x + 7 = 49 14x = 42 x = 3 4.

Steps

Original Equation Subtraction Property of Equality

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5. Solve the equation that follows. Justify each step in your process using the properties of equality. Be sure to include the properties of operations, if used.

8(2x – 1) = 56

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Summary… Identify the property represented below.

1.

x -3 = 6 x - 3 + 3 = 6 + 3 2. A = B, B = C, then A = C Solve the problem below justifying each step using the properties of equality.

3. 2x – 9 = 1

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Solving Equations with the Variable in Both Expressions of the Equation

1.

2.

3.

4.

5.

Move the

variable

equal sign. to solve for to the

left

of the Move

all other terms

to the right of the equal sign.

Combine

sign. like terms on

each side

of the equal Now solve for the

variable

and

simplify

.

Substitute

the solution into the

original

equation and

check

your work.

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Example 4: Solve the equation 5𝑥 + 9 = 2𝑥 − 36

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