Algebra Proofs Notes & Worksheets for High School

You're looking at a worksheet designed to teach Algebraic Proofs using a two-column format: Statements and Reasons. The goal is to logically justify each step taken to solve an equation, proving that the solution (e.g., x = 15) is correct.

Let’s solve and explain the first proof shown in the image:

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🎯 Problem:

Given: `2x - 10 = 20`
Prove: `x = 15`

We’ll fill in the Reasons column for each statement.

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## ✔ Step-by-Step Solution with Reasons

| Statements | Reasons |
|-------------------------------|------------------------------------------|
| 1. `2x - 10 = 20` | 1. Given — This is the starting equation provided in the problem. |
| 2. `2x - 10 + 10 = 20 + 10` | 2. Addition Property of Equality — We add 10 to both sides to isolate the term with x. This maintains equality. |
| 3. `2x = 30` | 3. Simplify / Combine Like Terms — On the left, `-10 + 10 = 0`, so we’re left with `2x`. On the right, `20 + 10 = 30`. |
| 4. `(2x)/2 = 30/2` | 4. Division Property of Equality — We divide both sides by 2 to solve for x. This keeps the equation balanced. |
| 5. `x = 15` | 5. Simplify — `2x ÷ 2 = x` and `30 ÷ 2 = 15`. |

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## 🧠 Explanation:

This is a classic two-column algebraic proof. Each step must be justified using a mathematical property or definition.

- “Given” means it’s the original information.
- Addition Property of Equality: If you add the same number to both sides of an equation, the equation remains true.
- Simplify: Combining numbers or terms to make the expression simpler.
- Division Property of Equality: If you divide both sides of an equation by the same nonzero number, the equation remains true.

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## 🔁 Why Do This?

In algebra, especially as you move toward geometry and higher math, showing *why* each step works is just as important as getting the right answer. This builds logical reasoning skills and prepares you for formal proofs later.

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## 💡 Bonus: Check Your Answer!

Always plug your solution back into the original equation to verify:

> Original: `2x - 10 = 20`
> Plug in `x = 15`:
> `2(15) - 10 = 30 - 10 = 20` ✔️
> It checks out!

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✔ So, the completed proof looks like this:

| Statements | Reasons |
|-------------------------------|------------------------------------------|
| 1. `2x - 10 = 20` | 1. Given |
| 2. `2x - 10 + 10 = 20 + 10` | 2. Addition Property of Equality |
| 3. `2x = 30` | 3. Simplify |
| 4. `(2x)/2 = 30/2` | 4. Division Property of Equality |
| 5. `x = 15` | 5. Simplify |

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Let me know if you’d like help with the other problems on the worksheet too! 😊