Algebra Proofs Notes and Worksheets - Lindsay Bowden - Free Printable

It looks like you've uploaded an image containing a geometry or algebraic reasoning worksheet with several problems involving two-column proofs (Statements and Reasons). However, the image itself isn't visible in this interface. But based on your description and the text content you've provided, I can help solve and explain each problem.

Let’s go through each section step by step.

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Problem 1:

Given:
- $ 2x + 40 = 30 + 10 $

We are to fill in the statements and reasons for a proof.

#### Step-by-step:

Statement 1: $ 2x + 40 = 30 + 10 $
Reason 1: Given

Statement 2: $ 2x + 40 = 40 $
Reason 2: Simplify (Addition)

Statement 3: $ 2x = 0 $
Reason 3: Subtract 40 from both sides (Subtraction Property of Equality)

Statement 4: $ x = 0 $
Reason 4: Divide both sides by 2 (Division Property of Equality)

✔ Final Answer: $ x = 0 $

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Problem 2:

Given: $ \frac{1}{2}a + 5 = 12 $, Prove: $ a = 14 $

#### Step-by-step:

Statement 1: $ \frac{1}{2}a + 5 = 12 $
Reason 1: Given

Statement 2: $ \frac{1}{2}a = 7 $
Reason 2: Subtract 5 from both sides (Subtraction Property of Equality)

Statement 3: $ a = 14 $
Reason 3: Multiply both sides by 2 (Multiplication Property of Equality)

✔ Final Answer: $ a = 14 $

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Problem 3:

Given: $ 4y + 3 = 1 $, Prove: $ y = -\frac{1}{2} $

#### Step-by-step:

Statement 1: $ 4y + 3 = 1 $
Reason 1: Given

Statement 2: $ 4y = -2 $
Reason 2: Subtract 3 from both sides (Subtraction Property of Equality)

Statement 3: $ y = -\frac{1}{2} $
Reason 3: Divide both sides by 4 (Division Property of Equality)

✔ Final Answer: $ y = -\frac{1}{2} $

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Problem 4:

Given: $ NQ = OJ $, $ OJ = 12 $, Prove: $ NQ = 12 $

This is a transitive property of equality.

#### Step-by-step:

Statement 1: $ NQ = OJ $
Reason 1: Given

Statement 2: $ OJ = 12 $
Reason 2: Given

Statement 3: $ NQ = 12 $
Reason 3: Transitive Property of Equality

✔ Final Answer: $ NQ = 12 $

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Summary of All Solutions:

| Problem | Given | Prove | Final Answer |
|--------|-------|------|--------------|
| 1 | $ 2x + 40 = 30 + 10 $ | $ x = ? $ | $ x = 0 $ |
| 2 | $ \frac{1}{2}a + 5 = 12 $ | $ a = ? $ | $ a = 14 $ |
| 3 | $ 4y + 3 = 1 $ | $ y = ? $ | $ y = -\frac{1}{2} $ |
| 4 | $ NQ = OJ $, $ OJ = 12 $ | $ NQ = ? $ | $ NQ = 12 $ |

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Key Concepts Used:

- Properties of Equality:
- Addition/Subtraction
- Multiplication/Division
- Transitive Property: If $ A = B $ and $ B = C $, then $ A = C $
- Simplifying expressions
- Two-column proof structure

If you have a specific question about any part, or if you'd like these formatted as a completed two-column proof table, let me know!