Question

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

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Answer 1

To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \), we will go through it step by step, applying the properties of equality as needed.

### Step 1: Distribute on the left side

We start by distributing \( 32 \) into the expression \( 32(4 - 8x) \):

\[
32 \cdot 4 - 32 \cdot 8x = 128 - 256x
\]

Now, we can rewrite the equation:

\[
128 - 256x + 5x = 1 - 7x
\]

### Step 2: Combine like terms on the left side

Next, combine the \( x \) terms on the left:

\[
128 - 256x + 5x = 128 - 251x
\]

Thus, we have:

\[
128 - 251x = 1 - 7x
\]

### Step 3: Isolate the \( x \) terms

To get all the \( x \) terms on one side and the constant terms on the other, we can add \( 251x \) to both sides:

\[
128 = 1 - 7x + 251x
\]

This simplifies to:

\[
128 = 1 + 244x
\]

### Step 4: Isolate the constant

Next, subtract \( 1 \) from both sides:

\[
128 - 1 = 244x
\]

This simplifies to:

\[
127 = 244x
\]

### Step 5: Solve for \( x \)

Now, divide both sides by \( 244 \):

\[
x = \frac{127}{244}
\]

### Step 6: Conclusion

We have found a single solution, \( x = \frac{127}{244} \).

### Number of Solutions

Since we derived a unique \( x \) value, we conclude that there is exactly **one solution** to the equation.

### Summary of Justifications

1. **Distributive Property**: Used to expand \( 32(4 - 8x) \).
2. **Combining Like Terms**: Combined \( -256x \) and \( 5x \) terms.
3. **Addition and Subtraction Properties of Equality**: Used to rearrange the equation by adding and subtracting terms on both sides.
4. **Division Property of Equality**: Divided both sides by \( 244 \) to isolate \( x \).

Thus, the final answer is:

\[
\text{There is exactly one solution: } x = \frac{127}{244}.
\]