Question

Solve the equation justifying each step with the correct reasoning.
2(x+8)=2x+8
step 1:______Property to get ________ simplified equation

Step 2: _____ Property to get ________ simplified equation

For this equation, there is/are ________

Properties and Reasons:
Subtraction of Equality
Combine Like Terms
Multiplication of Equality
Distributive
Division of Equality
Addition of Equality

Equation Simplified:
x+8+8, x+-8,
2x+8=8, Infinite Solutions,
No Solution, 16=8,
2x+16=2x+8, One solution,
2x+8=2x+8, x=2, x+16=8

Answer 1

Let's go through the equation step by step and justify each step with the correct reasoning.

The original equation is:

\[ 2(x + 8) = 2x + 8 \]

**Step 1:** Apply the **Distributive Property** to get the simplified equation:

\[ 2 \cdot x + 2 \cdot 8 = 2x + 8 \]

This simplifies to:

\[ 2x + 16 = 2x + 8 \]

**Step 2:** Now, we will apply the **Subtraction of Equality** property to isolate terms. We can subtract \(2x\) from both sides of the equation:

\[ 2x + 16 - 2x = 2x + 8 - 2x \]

This simplifies to:

\[ 16 = 8 \]

At this point, we see that 16 does not equal 8, indicating that there are **no solutions** for this equation.

Properties and Reasons:
- **Step 1:** Distributive Property to get \(2x + 16 = 2x + 8\)
- **Step 2:** Subtraction of Equality to get \(16 = 8\)

For this equation, there is/are **no solution**.

Final summary:
- We started with \(2(x + 8) = 2x + 8\).
- After applying the Distributive Property, we got \(2x + 16 = 2x + 8\).
- After applying the Subtraction of Equality, we arrived at \(16 = 8\), which is a contradiction (always false), thus indicating that there are no solutions to the equation.