Question

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

Answer 1

To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \) and determine the number of solutions, we will follow these steps, justifying each step using the properties of equality:

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

2. **Combine like terms on the left side**:
\[
128 - 251x = 1 - 7x
\]
- **Justification**: Combine like terms (i.e., \(-256x + 5x = -251x\))

3. **Add \( 251x \) to both sides to isolate \( x \)**:
\[
128 = 1 + 251x - 7x
\]
- **Justification**: Addition Property of Equality

4. **Combine like terms on the right side**:
\[
128 = 1 + 244x
\]
- **Justification**: Combine like terms (i.e., \( 251x - 7x = 244x \))

5. **Subtract 1 from both sides**:
\[
127 = 244x
\]
- **Justification**: Subtraction Property of Equality

6. **Divide both sides by 244**:
\[
x = \frac{127}{244}
\]
- **Justification**: Division Property of Equality

7. **Conclusion**:
The solution to the equation is \( x = \frac{127}{244} \), which means there is exactly **one solution** to the equation.

Thus, the number of solutions to the equation \( 32(4 - 8x) + 5x = 1 - 7x \) is **1**.