Algebra Proofs Notes and Worksheets - Lindsay Bowden - Free Printable

Here's the completed solution for each algebraic proof, with the correct Reasons filled in for each step. The goal is to justify each transformation using standard algebraic properties.

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Problem 1: Given: 2x - 10 = 20 → Prove: x = 15

| Statements | Reasons |
|-----------------------------|------------------------------------|
| 1. 2x - 10 = 20 | 1. Given |
| 2. 2x - 10 + 10 = 20 + 10 | 2. Addition Property of Equality (add 10 to both sides) |
| 3. 2x = 30 | 3. Simplify (or Arithmetic/Combining Like Terms) |
| 4. 2x/2 = 30/2 | 4. Division Property of Equality (divide both sides by 2) |
| 5. x = 15 | 5. Simplify |

✔ Explanation: We isolate `x` by first eliminating the constant term (-10) using addition, then eliminate the coefficient (2) using division.

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Problem 2: Given: x/2 + 5 = 12 → Prove: x = 14

| Statements | Reasons |
|-----------------------------|------------------------------------|
| 1. x/2 + 5 = 12 | 1. Given |
| 2. x/2 + 5 - 5 = 12 - 5 | 2. Subtraction Property of Equality (subtract 5 from both sides) |
| 3. x/2 = 7 | 3. Simplify |
| 4. (x/2) · 2 = 7 · 2 | 4. Multiplication Property of Equality (multiply both sides by 2) |
| 5. x = 14 | 5. Simplify |

✔ Explanation: First subtract 5 to isolate the fraction term, then multiply by 2 to cancel the denominator and solve for `x`.

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Problem 3: Given: 4(x + 3) = 16 → Prove: x = 1

| Statements | Reasons |
|-----------------------------|------------------------------------|
| 1. 4(x + 3) = 16 | 1. Given |
| 2. 4x + 12 = 16 | 2. Distributive Property |
| 3. 4x + 12 - 12 = 16 - 12 | 3. Subtraction Property of Equality (subtract 12 from both sides) |
| 4. 4x = 4 | 4. Simplify |
| 5. 4x/4 = 4/4 | 5. Division Property of Equality (divide both sides by 4) |
| 6. x = 1 | 6. Simplify |

✔ Explanation: Start by distributing the 4, then isolate the variable term by subtracting 12, and finally divide by 4 to solve for `x`.

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## ✔ Final Answer Summary:

Problem 1 Reasons:

1. Given
2. Addition Property of Equality
3. Simplify
4. Division Property of Equality
5. Simplify

Problem 2 Reasons:

1. Given
2. Subtraction Property of Equality
3. Simplify
4. Multiplication Property of Equality
5. Simplify

Problem 3 Reasons:

1. Given
2. Distributive Property
3. Subtraction Property of Equality
4. Simplify
5. Division Property of Equality
6. Simplify

These proofs demonstrate how to logically manipulate equations using fundamental algebraic properties to arrive at a proven solution.