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.