Starting Algebraic Proofs 2 Column | PDF | Geometry | Teaching ... - Free Printable

Let's solve the algebraic proof step by step and justify each step using the Algebraic Properties of Equality provided.

We are given:

> Given: $ 3x + 12 = 8x - 18 $
> Prove: $ x = 6 $

We'll fill in the Statements and Reasons table accordingly.

---

Step-by-step Solution:

#### Statement 1:
$ 3x + 12 = 8x - 18 $

Reason 1: Given

---

#### Statement 2:
$ 12 = 5x - 18 $

To get from Statement 1 to Statement 2, we subtract $ 3x $ from both sides:

$$
3x + 12 - 3x = 8x - 18 - 3x
\Rightarrow 12 = 5x - 18
$$

Reason 2: Subtraction Property of Equality
*(Subtracting $ 3x $ from both sides)*

---

#### Statement 3:
$ 30 = 5x $

Now, add 18 to both sides:

$$
12 + 18 = 5x - 18 + 18 \Rightarrow 30 = 5x
$$

Reason 3: Addition Property of Equality
*(Adding 18 to both sides)*

---

#### Statement 4:
$ 6 = x $

Divide both sides by 5:

$$
\frac{30}{5} = \frac{5x}{5} \Rightarrow 6 = x
$$

Reason 4: Division Property of Equality
*(Dividing both sides by 5)*

---

#### Statement 5:
$ x = 6 $

This is just rewriting $ 6 = x $ in standard form.

Reason 5: Symmetric Property of Equality
*(If $ 6 = x $, then $ x = 6 $)*

---

Final Answer Table:

| Statements | Reasons |
|--------------------------|----------------------------------------------|
| 1. $ 3x + 12 = 8x - 18 $ | 1. Given |
| 2. $ 12 = 5x - 18 $ | 2. Subtraction Property of Equality |
| 3. $ 30 = 5x $ | 3. Addition Property of Equality |
| 4. $ 6 = x $ | 4. Division Property of Equality |
| 5. $ x = 6 $ | 5. Symmetric Property of Equality |

---

✔ Conclusion: We have proven that $ x = 6 $ using valid algebraic properties at each step.