Practice worksheet for solving algebraic equations using two-column proofs, with space to write statements and reasons.

Let's solve each of the two-column algebraic proofs step by step, providing both statements and reasons for each. We'll go through all six problems.

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Problem 1:

Given: $-4x + 10 = -5x + 18$
Prove: $x = 8$

| Statements | Reasons |
|----------------|-------------|
| a. $-4x + 10 = -5x + 18$ | Given |
| b. $-4x + 10 + 5x = 18$ | Addition Property of Equality (add $5x$ to both sides) |
| c. $x + 10 = 18$ | Combine like terms |
| d. $x = 8$ | Subtraction Property of Equality (subtract 10 from both sides) |

> ✔ Final Answer: $x = 8$ ✔️

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Problem 2:

Given: $4x = 12x + 32$
Prove: $x = -4$

| Statements | Reasons |
|----------------|-------------|
| a. $4x = 12x + 32$ | Given |
| b. $4x - 12x = 32$ | Subtraction Property of Equality (subtract $12x$ from both sides) |
| c. $-8x = 32$ | Combine like terms |
| d. $x = -4$ | Division Property of Equality (divide both sides by $-8$) |

> ✔ Final Answer: $x = -4$ ✔️

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Problem 3:

Given: $\frac{3x + 5}{2} = 7$
Prove: $x = 3$

| Statements | Reasons |
|----------------|-------------|
| a. $\frac{3x + 5}{2} = 7$ | Given |
| b. $3x + 5 = 14$ | Multiplication Property of Equality (multiply both sides by 2) |
| c. $3x = 9$ | Subtraction Property of Equality (subtract 5 from both sides) |
| d. $x = 3$ | Division Property of Equality (divide both sides by 3) |

> ✔ Final Answer: $x = 3$ ✔️

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Problem 4:

Given: $5(x + 2) = -3x - 6$
Prove: $x = -2$

| Statements | Reasons |
|----------------|-------------|
| a. $5(x + 2) = -3x - 6$ | Given |
| b. $5x + 10 = -3x - 6$ | Distributive Property |
| c. $5x + 10 + 3x = -6$ | Addition Property of Equality (add $3x$ to both sides) |
| d. $8x + 10 = -6$ | Combine like terms |
| e. $8x = -16$ | Subtraction Property of Equality (subtract 10 from both sides) |
| f. $x = -2$ | Division Property of Equality (divide both sides by 8) |

> ✔ Final Answer: $x = -2$ ✔️

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Problem 5:

Given: $4 + 2(3x + 5) = 11 - x$
Prove: $x = -\frac{3}{7}$

| Statements | Reasons |
|----------------|-------------|
| a. $4 + 2(3x + 5) = 11 - x$ | Given |
| b. $4 + 6x + 10 = 11 - x$ | Distributive Property |
| c. $6x + 14 = 11 - x$ | Combine like terms |
| d. $6x + 14 + x = 11$ | Addition Property of Equality (add $x$ to both sides) |
| e. $7x + 14 = 11$ | Combine like terms |
| f. $7x = -3$ | Subtraction Property of Equality (subtract 14 from both sides) |
| g. $x = -\frac{3}{7}$ | Division Property of Equality (divide both sides by 7) |

> ✔ Final Answer: $x = -\frac{3}{7}$ ✔️

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Problem 6:

Given: $3 + 4(x - 2) = 27$
Prove: $x = 8$

| Statements | Reasons |
|----------------|-------------|
| a. $3 + 4(x - 2) = 27$ | Given |
| b. $3 + 4x - 8 = 27$ | Distributive Property |
| c. $4x - 5 = 27$ | Combine like terms |
| d. $4x = 32$ | Addition Property of Equality (add 5 to both sides) |
| e. $x = 8$ | Division Property of Equality (divide both sides by 4) |

> ✔ Final Answer: $x = 8$ ✔️

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✔ Summary of All Answers:

1. $x = 8$
2. $x = -4$
3. $x = 3$
4. $x = -2$
5. $x = -\frac{3}{7}$
6. $x = 8$

Each proof follows the two-column format, with statements on the left and reasons (properties of equality, distributive property, etc.) on the right.

Let me know if you'd like this formatted as a printable worksheet!