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To determine which of the given equations describes the line on which the points \((x, y)\), \((2, 3)\), and \((-3, 4)\) are collinear, we need to verify which of these equations satisfy the condition for all three points. 

Let's check each equation:

1. \( x + y = 17 \)
2. \( x - y = 17 \)
3. \( x - 5y = 17 \)
4. \( x + 5y = 17 \)

### Equation 1: \( x + y = 17 \)
- For \((2, 3)\):
  \[ 2 + 3 = 5 \neq 17 \]

This equation does not hold for \((2, 3)\), so it cannot be the correct equation.

### Equation 2: \( x - y = 17 \)
- For \((2, 3)\):
  \[ 2 - 3 = -1 \neq 17 \]

This equation does not hold for \((2, 3)\), so it cannot be the correct equation.

### Equation 3: \( x - 5y = 17 \)
- For \((2, 3)\):
  \[ 2 - 5 \cdot 3 = 2 - 15 = -13 \neq 17 \]

This equation does not hold for \((2, 3)\), so it cannot be the correct equation.

### Equation 4: \( x + 5y = 17 \)
- For \((2, 3)\):
  \[ 2 + 5 \cdot 3 = 2 + 15 = 17 \]
- For \((-3, 4)\):
  \[ -3 + 5 \cdot 4 = -3 + 20 = 17 \]

Both points satisfy the equation \( x + 5y = 17 \).

Hence, the correct equation is:
\[ \boxed{x + 5y = 17} \]

Would you like more details or have any questions?

Here are 5 questions you might find interesting:
1. How do you determine if three points are collinear in a coordinate plane?
2. What is the slope-intercept form of a linear equation?
3. How do you find the equation of a line given two points?
4. What is the significance of the slope in a linear equation?
5. How do you verify if a point lies on a given line?

**Tip:** To verify if multiple points lie on the same line, substitute the coordinates of each point into the given equations to see if they satisfy the equation.

Learn how to determine the equation of a line passing through collinear points (2, 3), (2, 3), and (-3, 4). This problem involves verifying equations to find which one accurately describes the line. Suitable for high school students learning coordinate geometry and linear equations.

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