Algebraic Proof Practice Worksheet with Properties of Equality

Actually, I can’t see or process images — including the one you uploaded — because I don’t have image-reading capabilities. However, based on your description and the text you’ve pasted, it looks like you’re working on algebraic proofs involving properties of equality (like Addition, Subtraction, Multiplication, Division, Symmetric, etc.).

Let’s go through each problem step-by-step using the text you provided.

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## ✔ Problem 1: Prove \( x = 5 \) given \( 2x - 3 = 7 \)

Given:

\( 2x - 3 = 7 \)

Proof:

| Statement | Reason |
|---------------------------|----------------------------------|
| 1. \( 2x - 3 = 7 \) | Given |
| 2. \( 2x = 10 \) | Addition Property of Equality (add 3 to both sides) |
| 3. \( x = 5 \) | Division Property of Equality (divide both sides by 2) |

✔ Conclusion: \( x = 5 \)

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## ✔ Problem 2: Prove \( x = 7 \) given \( \frac{x + 4}{3} = \frac{11}{3} \)

Given:

\( \frac{x + 4}{3} = \frac{11}{3} \)

Proof:

| Statement | Reason |
|------------------------------|----------------------------------------|
| 1. \( \frac{x + 4}{3} = \frac{11}{3} \) | Given |
| 2. \( x + 4 = 11 \) | Multiplication Property of Equality (multiply both sides by 3) |
| 3. \( x = 7 \) | Subtraction Property of Equality (subtract 4 from both sides) |

✔ Conclusion: \( x = 7 \)

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## ✔ Problem 3: Write an Algebraic Proof for the Converse of Column A → Column B

You wrote:

> Given: \( 2(x - 3) = 8 \) → Prove: \( x = 7 \)

Wait — actually, looking at your table, it seems you’re proving that if \( 2(x - 3) = 8 \), then \( x = 7 \). Let’s write a clean proof.

Given:

\( 2(x - 3) = 8 \)

Proof:

| Statement | Reason |
|--------------------------|-----------------------------------------|
| 1. \( 2(x - 3) = 8 \) | Given |
| 2. \( 2x - 6 = 8 \) | Distributive Property of Equality |
| 3. \( 2x = 14 \) | Addition Property of Equality (add 6 to both sides) |
| 4. \( x = 7 \) | Division Property of Equality (divide both sides by 2) |

✔ Conclusion: \( x = 7 \)

*(Note: The “converse” might be misleading here — this is just solving the equation algebraically.)*

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## ✔ Problem 4: Write an Algebraic Proof to show that if \( x = 4 \), then \( \frac{1}{2}x + 3 = 5 \)

This is a verification proof — we start with \( x = 4 \) and show that the expression equals 5.

Given:

\( x = 4 \)

Prove:

\( \frac{1}{2}x + 3 = 5 \)

Proof:

| Statement | Reason |
|-----------------------|-------------------------------------------|
| 1. \( x = 4 \) | Given |
| 2. \( \frac{1}{2}x = 2 \) | Multiplication Property of Equality (multiply both sides by ½) |
| 3. \( \frac{1}{2}x + 3 = 5 \) | Addition Property of Equality (add 3 to both sides) |

✔ Conclusion: \( \frac{1}{2}x + 3 = 5 \)

*(Alternatively, you could substitute directly: \( \frac{1}{2}(4) + 3 = 2 + 3 = 5 \))*

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## ✔ Problem 5: Write an Algebraic Proof to show that if \( x = 5 \), then \( 3x - 2 = 13 \)

Given:

\( x = 5 \)

Prove:

\( 3x - 2 = 13 \)

Proof:

| Statement | Reason |
|----------------------|-------------------------------------------|
| 1. \( x = 5 \) | Given |
| 2. \( 3x = 15 \) | Multiplication Property of Equality (×3) |
| 3. \( 3x - 2 = 13 \) | Subtraction Property of Equality (−2) |

✔ Conclusion: \( 3x - 2 = 13 \)

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## ✔ Problem 6: Write an Algebraic Proof to show that if \( x = 6 \), then \( \frac{x}{2} + 1 = 4 \)

Given:

\( x = 6 \)

Prove:

\( \frac{x}{2} + 1 = 4 \)

Proof:

| Statement | Reason |
|----------------------|-------------------------------------------|
| 1. \( x = 6 \) | Given |
| 2. \( \frac{x}{2} = 3 \) | Division Property of Equality (÷2) |
| 3. \( \frac{x}{2} + 1 = 4 \) | Addition Property of Equality (+1) |

✔ Conclusion: \( \frac{x}{2} + 1 = 4 \)

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## 🧠 Summary of Properties Used:

- Addition Property: If \( a = b \), then \( a + c = b + c \)
- Subtraction Property: If \( a = b \), then \( a - c = b - c \)
- Multiplication Property: If \( a = b \), then \( ac = bc \)
- Division Property: If \( a = b \) and \( c ≠ 0 \), then \( a/c = b/c \)
- Distributive Property: \( a(b + c) = ab + ac \)
- Symmetric Property: If \( a = b \), then \( b = a \) *(not used much in these proofs)*

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## 💡 Tip for Future Problems:

When writing algebraic proofs:
1. Start with what’s given.
2. Apply one property per step.
3. Clearly state the reason for each step.
4. End with what you’re trying to prove.

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If you want me to check your own handwritten answers or help format them into a neat table, feel free to type them out! I’m happy to review or correct them.

Let me know if you’d like to tackle more problems or need explanations on specific properties! 😊