The x-coordinates of the solutions to a system of equations are `3.5 and 6.` Which of the following could be the system ?

Options

A. `{{:(y =x +3.5),( y= x ^(2) +6):}`
B. `{{:(y =x -7),( y=- (x -6)^(2)):}`
C. `{{:(y =1/2 x +3),(y =- (x -5) ^(2) +7):}`
D. `{{:(y =1/2 x +7 ),(y=- (x-6) ^(2) + 3.5):}`

Explanation

To solve the problem of identifying which system of equations has solutions with x-coordinates of 3.5 and 6, we will evaluate each option provided in the question. We will substitute the x-coordinates into the equations and check if the corresponding y-values are equal for both equations in each option.

### Step-by-Step Solution:

1. **Evaluate Option A:**
- First equation: \( y = x + 3.5 \)
- Substitute \( x = 6 \):
\[ y = 6 + 3.5 = 9.5 \]
- Second equation: \( y = x^2 + 6 \)
- Substitute \( x = 6 \):
\[ y = 6^2 + 6 = 36 + 6 = 42 \]
- Since \( 9.5 \neq 42 \), Option A is not correct.

2. **Evaluate Option B:**
- First equation: \( y = 2x - 7 \)
- Substitute \( x = 6 \):
\[ y = 2(6) - 7 = 12 - 7 = 5 \]
- Second equation: \( y = - (x - 6)^2 \)
- Substitute \( x = 6 \):
\[ y = - (6 - 6)^2 = -0^2 = 0 \]
- Since \( 5 \neq 0 \), Option B is not correct.

3. **Evaluate Option C:**
- First equation: \( y = \frac{1}{2}x + 3 \)
- Substitute \( x = 6 \):
\[ y = \frac{1}{2}(6) + 3 = 3 + 3 = 6 \]
- Second equation: \( y = - (x - 5)^2 + 7 \)
- Substitute \( x = 6 \):
\[ y = - (6 - 5)^2 + 7 = -1^2 + 7 = -1 + 7 = 6 \]
- Since \( 6 = 6 \), Option C is a potential correct answer.

4. **Evaluate Option D:**
- First equation: \( y = \frac{1}{2}x + 7 \)
- Substitute \( x = 6 \):
\[ y = \frac{1}{2}(6) + 7 = 3 + 7 = 10 \]
- Second equation: \( y = - (x - 6)^2 + 3.5 \)
- Substitute \( x = 6 \):
\[ y = - (6 - 6)^2 + 3.5 = -0^2 + 3.5 = 0 + 3.5 = 3.5 \]
- Since \( 10 \neq 3.5 \), Option D is not correct.

### Conclusion:
The only option where both equations yield the same y-value for the x-coordinates of 3.5 and 6 is **Option C**.