A line through the origin intersects the parabola `5y=2x^(2)-9x+10` at two points whose x-coordinates add up to 17.
Then the slope of the line is __________ .

Explanation

To solve the problem, we need to find the slope of a line that intersects the given parabola at two points, where the sum of the x-coordinates of these intersection points is 17.

### Step-by-Step Solution:

1. **Equation of the Line**:
Since the line passes through the origin, we can express it as:
\[ y = mx \]
where \( m \) is the slope of the line.

2. **Substituting into the Parabola**:
The equation of the parabola is given as:
\[ 5y = 2x^2 - 9x + 10 \]
Substitute \( y = mx \) into the parabola's equation:
\[ 5(mx) = 2x^2 - 9x + 10 \]
This simplifies to:
\[ 5mx = 2x^2 - 9x + 10 \]

3. **Rearranging the Equation**:
Rearranging gives us:
\[ 2x^2 - (9 + 5m)x + 10 = 0 \]
This is a quadratic equation in \( x \).

4. **Sum of Roots**:
For a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots is given by:
\[ -\frac{b}{a} \]
Here, \( a = 2 \) and \( b = -(9 + 5m) \). Therefore, the sum of the roots is:
\[ -\frac{-(9 + 5m)}{2} = \frac{9 + 5m}{2} \]

5. **Setting the Sum of Roots Equal to 17**:
According to the problem, the sum of the x-coordinates of the intersection points is 17:
\[ \frac{9 + 5m}{2} = 17 \]

6. **Solving for \( m \)**:
Multiply both sides by 2:
\[ 9 + 5m = 34 \]
Subtract 9 from both sides:
\[ 5m = 25 \]
Divide by 5:
\[ m = 5 \]

### Final Answer:
The slope of the line is \( \boxed{5} \).