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)

Answer 1

To solve the equation \( 32(4 - 8x) + 5x = 1 - 7x \), we will follow a series of steps, justifying each one using properties of equality.

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

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

3. **Add** \( 251x \) to both sides:
\[
128 = 1 - 7x + 251x
\]
\[
128 = 1 + 244x
\]
(Justification: Addition Property of Equality)

4. **Subtract** \( 1 \) from both sides:
\[
128 - 1 = 244x
\]
\[
127 = 244x
\]
(Justification: Addition Property of Equality)

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

Now that we have solved for \( x \), we can conclude that:

- The equation has **one solution**, which is \( x = \frac{127}{244} \).

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