Algebra Proofs Notes and Worksheets - Lindsay Bowden - Free Printable

Problem: Algebra Proofs Practice

We are tasked with completing the algebra proofs by filling in the reasons for each step. Let's solve each problem step by step.

---

#### 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 |
| 4. \(\frac{2x}{2} = \frac{30}{2}\) | 4. Division Property of Equality (Divide both sides by 2) |
| 5. \(x = 15\) | 5. Simplification |

---

#### 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. Simplification |
| 4. \(\frac{x}{2} \cdot 2 = 7 \cdot 2\) | 4. Multiplication Property of Equality (Multiply both sides by 2) |
| 5. \(x = 14\) | 5. Simplification |

---

#### 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. Simplification |
| 5. \(\frac{4x}{4} = \frac{4}{4}\) | 5. Division Property of Equality (Divide both sides by 4) |
| 6. \(x = 1\) | 6. Simplification |

---

Final Answer:

\[
\boxed{
\begin{array}{|c|c|}
\hline
\text{Problem 1} & \\
\hline
\text{Statements} & \text{Reasons} \\
\hline
1. 2x - 10 = 20 & 1. \text{Given} \\
2. 2x - 10 + 10 = 20 + 10 & 2. \text{Addition Property of Equality} \\
3. 2x = 30 & 3. \text{Simplification} \\
4. \frac{2x}{2} = \frac{30}{2} & 4. \text{Division Property of Equality} \\
5. x = 15 & 5. \text{Simplification} \\
\hline
\text{Problem 2} & \\
\hline
\text{Statements} & \text{Reasons} \\
\hline
1. \frac{x}{2} + 5 = 12 & 1. \text{Given} \\
2. \frac{x}{2} + 5 - 5 = 12 - 5 & 2. \text{Subtraction Property of Equality} \\
3. \frac{x}{2} = 7 & 3. \text{Simplification} \\
4. \frac{x}{2} \cdot 2 = 7 \cdot 2 & 4. \text{Multiplication Property of Equality} \\
5. x = 14 & 5. \text{Simplification} \\
\hline
\text{Problem 3} & \\
\hline
\text{Statements} & \text{Reasons} \\
\hline
1. 4(x + 3) = 16 & 1. \text{Given} \\
2. 4x + 12 = 16 & 2. \text{Distributive Property} \\
3. 4x + 12 - 12 = 16 - 12 & 3. \text{Subtraction Property of Equality} \\
4. 4x = 4 & 4. \text{Simplification} \\
5. \frac{4x}{4} = \frac{4}{4} & 5. \text{Division Property of Equality} \\
6. x = 1 & 6. \text{Simplification} \\
\hline
\end{array}
}
\]