1

Lesson 2.1 Solving Equations w/Justification

Concept: Solving Equations

EQ: How do we justify how we solve equations? REI. 1

Vocabulary:Properties of EqualityProperties of OperationJustify

2

Solve the equations below, provide an explanation for your steps.1. 2x – 3 = 13

Properties of Equality

2.1.1: Properties of Equality

3

Property In symbols Example

Reflexive propertyof equality

a = a 2=2

Symmetric propertyof equality

If a = b, then b = a.

x = 33 = x

Transitive propertyof equality

If a = b and b = c, then a = c.

x = 2, y = 2, x = y

Addition propertyof equality

If a = b, then a + c = b + c.

x – 4 = 3x – 4 + 4 = 3 + 4

x = 7

Properties of Equality, continued

2.1.1: Properties of Equality

4

Property In symbols Examples

Subtractionproperty of equality

If a = b, then a – c = b – c.

x + 2 =5x + 2 – 2 = 5 – 2

x = 3

Multiplicationproperty of equality

If a = b and c ≠ 0, thena • c = b • c. x=15

Division propertyof equality

If a = b and c ≠ 0, then a ÷ c = b ÷ c.

4x = 16

x = 4

Properties of Equality, continued

2.1.1: Properties of Equality

5

Property In symbols Examples

Substitutionproperty of equality

If a = b, then b may besubstituted for a in anyexpression containing a.

x = 3, then2x = 2(3) = 6

Properties of Operations

2.1.1: Properties of Equality

6

Property General rule Specific exampleCommutative property of addition

a + b = b + a 3 + 8 = 8 + 3

Associative property of addition

(a + b) + c = a + (b + c)

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

Commutative property ofmultiplication

a • b = b • a 3 • 8 = 8 • 3

Associative property ofmultiplication

(a • b) • c = a • (b • c) (3 • 8) • 2 = 3 • (8 • 2)

Distributive property ofmultiplication over addition

a • (b + c) = a • b + a • c

3 • (8 + 2) = 3 • 8 + 3 • 2

Guided Practice

Example 1Which property of equality is missing in the steps to solve the equation –7x + 22 = 50?

2.1.1: Properties of Equality

7

Equation Steps–7x + 22 = 50 Original equation

–7x = 28

x = –4 Division property of equality

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.

2.1.1: Properties of Equality

8

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.”

2.1.1: Properties of Equality

9

✔

Guided Practice: Example 1, continued

2.1.1: Properties of Equality

10

Guided Practice

Example 2Which property of equality is missing in the steps to

solve the equation

2.1.1: Properties of Equality

11

Equation Steps

Original equation

Addition property of equality

–x = 42

x = –42 Division property of equality

−𝑥6

=7

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,

and 4. The result of this step is .

2.1.1: Properties of Equality

12

Guided Practice: Example 2, continuedIn 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.

The result of multiplying by 6 is –x and

the result of multiplying 7 by 6 is 42. This

matches the next step in the sequence.

2.1.1: Properties of Equality

13

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.”

2.1.1: Properties of Equality

14

✔

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 3rd and 5th equations.

2. Read the 4th 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.

3.

4.

17

Equation Steps

9 + x = 17 Original Equation

x = 8

Equation Steps

7(2x + 1) = 49 Original Equation

14x + 7 = 49

14x = 42 Subtraction Property of Equality

x = 3

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

18

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

19

Solving Equations with the Variable in Both Expressions of the Equation

1. Move the variable to solve for to the left of the equal sign.

2. Move all other terms to the right of the equal sign.

3. Combine like terms on each side of the equal sign.

4. Now solve for the variable and simplify.

5. Substitute the solution into the original equation and check your work.

20

Example 4: Solve the equation

21