Geometry Worksheet - Algebraic Proof by Word of Math worksheets library - Free Printable

Let's solve each algebraic proof step by step and fill in the Reasons column appropriately. The goal is to justify each step using 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 (combine like terms) |
| 4. $\frac{2x}{2} = \frac{30}{2}$ | 4. Division Property of Equality (divide both sides by 2) |
| 5. $x = 15$ | 5. Simplify |

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Problem 2:

Given: $\frac{x}{2} + 5 = 12$
Prove: $x = 14$

| Statements | Reasons |
|-----------|--------|
| 1. $\frac{x}{2} + 5 = 12$ | 1. Given |
| 2. $\frac{x}{2} + 5 - 5 = 12 - 5$ | 2. Subtraction Property of Equality (subtract 5 from both sides) |
| 3. $\frac{x}{2} = 7$ | 3. Simplify |
| 4. $\frac{x}{2} \cdot 2 = 7 \cdot 2$ | 4. Multiplication Property of Equality (multiply both sides by 2) |
| 5. $x = 14$ | 5. Simplify |

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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) |
| 3. $4x + 12 - 12 = 16 - 12$ | 3. Subtraction Property of Equality (subtract 12 from both sides) |
| 4. $4x = 4$ | 4. Simplify |
| 5. $\frac{4x}{4} = \frac{4}{4}$ | 5. Division Property of Equality (divide both sides by 4) |
| 6. $x = 1$ | 6. Simplify |

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✔ Summary of Key Properties Used:

- Given: The starting equation.
- Addition/Subtraction Property of Equality: You can add or subtract the same value from both sides of an equation.
- Multiplication/Division Property of Equality: You can multiply or divide both sides by the same non-zero number.
- Distributive Property: $a(b + c) = ab + ac$
- Simplify: Combine like terms or reduce fractions.

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These proofs follow a logical sequence, justifying each step with a mathematical property. This format helps build strong reasoning skills in algebra.