Question

A student solved the equation: x + 2 (x + 1) = 17
Line 1: x + 2x + 1 = 17
Line 2: 3x + 1 = 17
Line 3: 3x = 16
Line 4: x = 16/3
Identify the Property used to go from Line 2 to Line 3: Distributive, Multiplication property of equality, or Subtraction Property of Equality?
Which line has a mistake?
What should have been the correct answer for the equation: No solution, x = 5, x = 45, x = 15

Answer 1

Let's analyze the student's solution step by step:

1. **Line 1:** \( x + 2(x + 1) = 17 \) becomes \( x + 2x + 2 = 17 \) (not 1).
- Here is the mistake. The student incorrectly expanded \( 2(x + 1) \). The correct expansion should be \( 2x + 2 \).

2. **Line 2:** \( 3x + 1 = 17 \) is derived from Line 1, and it should actually be \( 3x + 2 = 17 \).

3. **Line 3:** The simplification to \( 3x = 16 \) is incorrect due to the previous mistake. The correct equation should have been \( 3x + 2 = 17\), leading to \( 3x = 15 \).

4. **Line 4:** Claims \( x = \frac{16}{3} \) is therefore incorrect based on the previous mistakes.

To answer your questions:

1. **Identify the Property used to go from Line 2 to Line 3:** This should actually involve the **Subtraction Property of Equality**, but in this case, since Line 2 itself is incorrect, this property was not applied correctly.

2. **Which line has a mistake?** Line 1 has a mistake in the expansion.

3. **What should have been the correct answer for the equation?**
- Following the correct derivation: From \( 3x + 2 = 17 \):
- Subtract 2 from both sides:
- \( 3x = 15 \)
- Divide by 3:
- \( x = 5 \)
- Therefore, the correct answer is **x = 5**.