1. Simplifying Expressions and SOLVING Equations
Using TWO-COLUMN Proofs

2. Simplify the expression below.

3. Simplify the expression below.
Normally, we would simply multiply 9 and 4
Then multiply the product by 5

4. Simplify the expression below.
Now, we will use TWO-COLUMN Proofs
Step 1: Make 2 Columns
Step 2: Label left-column STATEMENTS
Step 3: Label right-column REASONS
STATEMENTS
REASONS

5. Simplify the expression below.
Now, we will use TWO-COLUMN Proofs
Step 4: Write the GIVEN information in
the upper left-hand column
Step 5: Write the reason for this: GIVEN
STATEMENTS
REASONS
1. 5(94)
Given

6. Simplify the expression below.
Now, we will use TWO-COLUMN Proofs
Step 6: Complete the table with correct
statements and reasons

7. Simplify the expression below.
The TWO-COLUMN Proof
STATEMENTS
REASONS
1. 5(94)
Given
2. 5(49)
2. Commutative Property of ✕
3. (54)9
3. Associative Property of ✕
4. 209
4. Substitution Property of =
180
5. Substitution Property of =

8. Simplify the expression below.
The TWO-COLUMN Proof
"quod erat demonstrandum"
("that which was to be demonstrated”)
Always ends a Proof.

9. Simplify the expression below.
The TWO-COLUMN Proof
STATEMENTS
REASONS
1. (33+288)+367 1.
2. (288+33)+367 2.
(33+367) 3.
(400) 4.
688 5.

10. Simplify the expression below.
The TWO-COLUMN Proof
STATEMENTS
REASONS
1. 3(2x – 4) + 2x 1.
2. 6x – x 2.
3. 6x + 2x – 12 3.
4. 8x – 12 4.

11. Simplify the expression below.
The TWO-COLUMN Proof
STATEMENTS
REASONS
1. -2(8x – 4) + 9x + 6 1.
2. -16x x + 6 2.
3. -16x + 9x 3.
4. -7x + 14 4.

12. Prove the solution for Equation below is y = -2
STATEMENTS
REASONS
1. 2y – 6 = -10 1.
2. 2y – = 2.
3. 2y + 0 = 3.
4. 2y = 4.
5. 2y = -4 5. 6.
7. y = -2 7.

13. Prove the solution for Equation below is y = -2
STATEMENTS
REASONS
1. 2y – 6 = -10
Given
2. 2y – =
2. Addition Property of =
3. 2y + 0 =
3. Additive Inverse Property
4. 2y =
4. Additive Identity Property
5. 2y = -4
5. Substitution Property of = 6.
6. Division Property of =
7. y = -2
7. Substitution Property of =

14. Prove the solution for Equation below is x = 10
STATEMENTS
REASONS
= 2(x – 4) 1.
2. 12 = 2x – 8 2.
= 2x – 8 + 8 3.
= 2x + 0 4.
= 2x 5.
6. 20 = 2x 6. 7.
8. 10 = x 8.
9. x = 10 9.

15. Prove the solution for Equation below is x = 10
STATEMENTS
REASONS
= 2(x – 4)
Given
2. 12 = 2x – 8
2. Distributive Property
= 2x – 8 + 8
3. Addition Property of =
= 2x + 0
4. Additive Inverse Property
= 2x
5. Additive Identity Property
6. 20 = 2x
6. Substitution Property of = 7.
7. Division Property of =
8. 10 = x
8. Substitution Property of =
9. x = 10
9. Symmetric Property of =