Justify your steps using the properties of equality and determine the number of solutions to the equation 32(4−8x)+5x=1−7x .

To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \) and justify the steps using the properties of equality, let’s break it down step by step.

1. **Distribute on the left side:**
\[
32(4) - 32(8x) + 5x = 1 - 7x
\]
\[
128 - 256x + 5x = 1 - 7x
\]

2. **Combine like terms on the left side:**
\[
128 - 251x = 1 - 7x
\]

3. **Isolate the variable \(x\):**
We want to get all \(x\) terms on one side and constant terms on the other side. Let's add \(7x\) to both sides:
\[
128 - 251x + 7x = 1
\]
This simplifies to:
\[
128 - 244x = 1
\]

Now, subtract \(128\) from both sides:
\[
-244x = 1 - 128
\]
\[
-244x = -127
\]

4. **Solve for \(x\):**
Now divide both sides by \(-244\):
\[
x = \frac{-127}{-244} = \frac{127}{244}
\]

5. **Conclusion:**
We found one unique solution for \(x\), which is \(x = \frac{127}{244}\).

Therefore, the equation \( 32(4 - 8x) + 5x = 1 - 7x \) has exactly **one solution**.

Can you give me in a sentence

The equation \( 32(4 - 8x) + 5x = 1 - 7x \) has exactly one solution, \( x = \frac{127}{244} \).

Is this right

Yes, my previous solution was correct. The equation \( 32(4 - 8x) + 5x = 1 - 7x \) simplifies to give a unique solution for \(x\), which is \(x = \frac{127}{244}\).

If you need further clarification or want to verify the steps, please let me know!

Yes

Let's verify the solution step-by-step for the equation \( 32(4 - 8x) + 5x = 1 - 7x \).

### Original Equation
\[ 32(4 - 8x) + 5x = 1 - 7x \]

### Step 1: Distribute on the Left Side
Distributing \(32\) gives us:
\[
32 \cdot 4 - 32 \cdot 8x + 5x = 1 - 7x
\]
\[
128 - 256x + 5x = 1 - 7x
\]

### Step 2: Combine Like Terms
Combine the \(x\) terms on the left side:
\[
128 - 251x = 1 - 7x
\]

### Step 3: Add \(7x\) to Both Sides
\[
128 - 251x + 7x = 1
\]
\[
128 - 244x = 1
\]

### Step 4: Isolate \(x\)
Subtract \(128\) from both sides:
\[
-244x = 1 - 128
\]
\[
-244x = -127
\]

### Step 5: Divide by \(-244\)
\[
x = \frac{-127}{-244}
\]
\[
x = \frac{127}{244}
\]

### Conclusion
Thus, the equation has one unique solution:
\[ x = \frac{127}{244} \]

We can verify this solution by substituting \(x = \frac{127}{244}\) back into the original equation to confirm that both sides are equal.

If you would like to see the verification step or have any further questions, please let me know!