https://file.aiquickdraw.com/mathbot/file/1727453116290vwr2eq24.jpg

Let's analyze the given equations to determine which ones have exactly one solution. The key idea is that for an equation to have exactly one solution, it should simplify to a form where the variable \( x \) can be isolated to a single unique value.

The equations in the image are:

- A: \(-19x + 18 = -19x + 18\)
- B: \(-19x - 18 = -19x + 18\)
- C: \(19x + 18 = -19x + 18\)
- D: \(19x - 18 = -19x + 18\)

Now, we'll simplify each equation:

### Equation A:
\[
-19x + 18 = -19x + 18
\]
Both sides are identical, so this equation holds true for all values of \( x \). This is an example of an equation with **infinite solutions**.

### Equation B:
\[
-19x - 18 = -19x + 18
\]
Simplify:
\[
-18 \neq 18
\]
This is a contradiction, meaning there is **no solution** to this equation.

### Equation C:
\[
19x + 18 = -19x + 18
\]
Move all terms involving \( x \) to one side:
\[
19x + 19x = 18 - 18
\]
\[
38x = 0
\]
\[
x = 0
\]
This equation has exactly **one solution**, \( x = 0 \).

### Equation D:
\[
19x - 18 = -19x + 18
\]
Move all terms involving \( x \) to one side:
\[
19x + 19x = 18 + 18
\]
\[
38x = 36
\]
\[
x = \frac{36}{38} = \frac{18}{19}
\]
This equation has exactly **one solution**, \( x = \frac{18}{19} \).

### Conclusion:
The equations with exactly one solution are **C** and **D**.

Do you want more details on how the solutions were derived? 

Here are 5 questions you could explore next:
1. What is the difference between an equation having no solution, one solution, or infinite solutions?
2. How does the distributive property affect solving these types of equations?
3. Can you create an equation similar to example D that has a different solution?
4. How do systems of equations relate to solving individual linear equations like these?
5. What happens if you have two equations that contradict each other in a system?

**Tip:** Always try to isolate the variable on one side of the equation to easily determine the number of solutions.

Which of the following equations have exactly one solution?

Understand which of the given linear equations have exactly one solution. This problem involves isolating variables and recognizing contradictions, providing a step-by-step solution suitable for students in Grades 7-10.

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