Repeat the following procedure for the four given numbers.
Multiply the number by 6. Add 4 to the product. Divide this sum by 2. Subtract 2 from the quotient.
The 1st number is 1.
The result is
The 2nd number is 4.
The result is
The 3rd number is 7.
The result is
The 4th number is 11.
The result is
a. Write a conjecture that relates the result of the process to the original number selected. Represent the original number as n.
The result is
(Simplify your answer.)
b. Represent the original number as n, and use deductive reasoning to prove the conjecture ir-part (a).
Multiply the number by 6.
Add 4 to the product.
Divide the sum by 2.
(Simplify your answer.)
Subtract 2 from the quotient.

Let's follow the procedure step-by-step for each number and then analyze the results.

### Procedure:

1. **Multiply the number by 6.**
2. **Add 4 to the product.**
3. **Divide this sum by 2.**
4. **Subtract 2 from the quotient.**

### Step-by-step calculations:

#### 1st Number: \( 1 \)

1. Multiply by 6: \( 1 \times 6 = 6 \)
2. Add 4: \( 6 + 4 = 10 \)
3. Divide by 2: \( 10 \div 2 = 5 \)
4. Subtract 2: \( 5 - 2 = 3 \)

The result is **3**.

#### 2nd Number: \( 4 \)

1. Multiply by 6: \( 4 \times 6 = 24 \)
2. Add 4: \( 24 + 4 = 28 \)
3. Divide by 2: \( 28 \div 2 = 14 \)
4. Subtract 2: \( 14 - 2 = 12 \)

The result is **12**.

#### 3rd Number: \( 7 \)

1. Multiply by 6: \( 7 \times 6 = 42 \)
2. Add 4: \( 42 + 4 = 46 \)
3. Divide by 2: \( 46 \div 2 = 23 \)
4. Subtract 2: \( 23 - 2 = 21 \)

The result is **21**.

#### 4th Number: \( 11 \)

1. Multiply by 6: \( 11 \times 6 = 66 \)
2. Add 4: \( 66 + 4 = 70 \)
3. Divide by 2: \( 70 \div 2 = 35 \)
4. Subtract 2: \( 35 - 2 = 33 \)

The result is **33**.

---

### Part (a): Write a Conjecture

By looking at the results for each number, we notice that the result of the process appears to be:

\[ \text{Result} = 3n \]

Where \( n \) is the original number. Let's prove this in part (b).

---

### Part (b): Prove the Conjecture

We need to prove that the result after the procedure is always \( 3n \).

Let the original number be \( n \).

#### Step-by-step process:

1. **Multiply by 6**: \( 6n \)
2. **Add 4**: \( 6n + 4 \)
3. **Divide by 2**: 
   \[
   \frac{6n + 4}{2} = 3n + 2
   \]
4. **Subtract 2**: 
   \[
   (3n + 2) - 2 = 3n
   \]

Thus, after following the procedure, the result is indeed \( 3n \), which proves the conjecture.

---

### Summary

The result of the procedure is always \( 3n \), where \( n \) is the original number selected.

---

Would you like further details or have any questions?

Here are 5 relative questions based on this:

1. What happens if the original number \( n \) is negative?
2. What if the original number \( n \) is a fraction?
3. Can you modify the procedure to produce a different linear relationship, say \( 4n \)?
4. How does the final result compare when you start with an even number vs. an odd number?
5. Is there a way to reverse the process to retrieve the original number?

**Tip:** Always check if there's a pattern when performing multiple similar calculations—this can help in forming conjectures!

Repeat the following procedure for the four given numbers. Multiply the number by 6. Add 4 to the product. Divide this sum by 2. Subtract 2 from the quotient. The 1st number is 1. The result is The 2nd number is 4. The result is The 3rd number is 7. The result is The 4th number is 11. The result is a. Write a conjecture that relates the result of the process to the original number selected. Represent the original number as n. The result is (Simplify your answer.) b. Represent the original number as n, and use deductive reasoning to prove the conjecture in part (a). Multiply the number by 6. Add 4 to the product. Divide the sum by 2. (Simplify your answer.) Subtract 2 from the quotient.

Learn how to solve a linear algebra problem by following a step-by-step process: multiply by 6, add 4, divide by 2, and subtract 2. This example leads to a conjecture (Result = 3n) with a deductive proof. Suitable for Grades 7-9.

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