Algebra Proofs Notes and Worksheets - Lindsay Bowden - Free Printable

Here's the completed solution with reasons filled in for each step of the algebraic proofs. The goal is to justify each transformation using standard algebraic properties or axioms.

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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. Simplification (or Arithmetic) |
| 4. 2x/2 = 30/2 | 4. Division Property of Equality (divide both sides by 2) |
| 5. x = 15 | 5. Simplification (or Arithmetic) |

✔ Explanation: We isolate `x` by first undoing subtraction (add 10), then undoing multiplication by 2 (divide by 2).

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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. Simplification |
| 4. (x/2) · 2 = 7 · 2 | 4. Multiplication Property of Equality (multiply both sides by 2) |
| 5. x = 14 | 5. Simplification |

✔ Explanation: First, remove the constant (+5) by subtracting 5. Then, eliminate the denominator by multiplying both sides by 2.

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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 (distribute 4 over x + 3) |
| 3. 4x + 12 - 12 = 16 - 12 | 3. Subtraction Property of Equality (subtract 12 from both sides) |
| 4. 4x = 4 | 4. Simplification |
| 5. 4x/4 = 4/4 | 5. Division Property of Equality (divide both sides by 4) |
| 6. x = 1 | 6. Simplification |

✔ 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. Simplification
4. Division Property of Equality
5. Simplification

Problem 2 Reasons:
1. Given
2. Subtraction Property of Equality
3. Simplification
4. Multiplication Property of Equality
5. Simplification

Problem 3 Reasons:
1. Given
2. Distributive Property
3. Subtraction Property of Equality
4. Simplification
5. Division Property of Equality
6. Simplification

These proofs demonstrate how to logically transform an equation step-by-step using algebraic properties to isolate the variable and prove the solution.