https://r2.erweima.ai/mathbot/image/1726421295427amoppv0b.jpg

We are given two equations from the image:

1. \( x - y = y - 4 + 2(4.5 - 2x) \)
2. \( 4(x + 2y) = -p + 7y \)

The goal is to find the value of \( p \) for which there is exactly one solution \( (x, y) \) when \( x = -1 \).

### Step 1: Simplify the first equation.

Start with the first equation:
\[
x - y = y - 4 + 2(4.5 - 2x)
\]
Expand and simplify the right-hand side:
\[
2(4.5 - 2x) = 9 - 4x
\]
Thus, the equation becomes:
\[
x - y = y - 4 + 9 - 4x
\]
Simplify further:
\[
x - y = y + 5 - 4x
\]
Now collect like terms:
\[
x - y = y + 5 - 4x \quad \Rightarrow \quad x + 4x - y = y + 5 \quad \Rightarrow \quad 5x - y = y + 5
\]
Combine like terms:
\[
5x = 2y + 5
\]
Therefore, the simplified first equation is:
\[
5x = 2y + 5
\]

### Step 2: Solve the second equation.

Now, take the second equation:
\[
4(x + 2y) = -p + 7y
\]
Expand:
\[
4x + 8y = -p + 7y
\]
Solve for \( p \):
\[
p = -4x + y
\]

### Step 3: Substitute \( x = -1 \) and find \( p \).

From the first equation, substitute \( x = -1 \):
\[
5(-1) = 2y + 5
\]
Simplify:
\[
-5 = 2y + 5 \quad \Rightarrow \quad 2y = -10 \quad \Rightarrow \quad y = -5
\]

Now, substitute \( x = -1 \) and \( y = -5 \) into the equation for \( p \):
\[
p = -4(-1) + (-5)
\]
Simplify:
\[
p = 4 - 5 = -1
\]

### Conclusion:
There is no answer that directly matches \( p = -1 \). Therefore, the correct choice is:
\[
\boxed{D} \text{ None of the above}
\]

Would you like further details or clarification?

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Here are five related questions for further exploration:

1. What conditions ensure a system of linear equations has exactly one solution?
2. How can you determine if a system of equations is consistent and independent?
3. What happens if both equations in a system are dependent (same line)?
4. How would you solve the system graphically?
5. How do changes in the value of \( p \) affect the number of solutions?

**Tip**: Always simplify the equations step-by-step to avoid errors, especially when constants are involved.

x - y = y - 4 + 2(4.5 - 2x), 4(x + 2y) = -p + 7y. In the system of equations, p is a constant. For which value of p is there exactly one solution (x, y) where x = -1?

Learn how to solve a system of equations to find the constant p when x = -1. This problem involves algebraic manipulation of linear equations and substitution. Suitable for students in Grades 9-11.

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