We are asked to find the gradient (slope) and the equation of the line passing through two points: (3, 5) and (-1, 9). Let's break this down step by step.

Step 1: Find the slope (m)

The formula for the slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points (3, 5) and (-1, 9): $m = \frac{9 - 5}{-1 - 3} = \frac{4}{-4} = -1$

Step 2: Use point-slope form to find the equation of the line

The point-slope form of a line is:

$y - y_1 = m(x - x_1)$

Using the slope $m = -1$ and the point (3, 5): $y - 5 = -1(x - 3)$ Simplifying: $y - 5 = -x + 3$ $y = -x + 3 + 5$ $y = -x + 8$

Step 3: Convert to standard form

The standard form of a line is usually written as $Ax + By = C$. We can rearrange the equation:

$x + y = 8$

To match the forms in the answer choices, let's rewrite it as: $x + y - 8 = 0 \quad \text{or} \quad x + y = 8$

However, none of the answer choices exactly match this form. Let's recheck the options and ensure the calculation aligns with the problem's specific instructions.

Would you like me to double-check or clarify any part of the problem further? Here are some related questions:

1. How do we derive the equation of a line from two points?
2. What are the different forms of a line equation (slope-intercept, point-slope, and standard)?
3. How can we interpret the slope in the context of geometry?
4. How do we convert between slope-intercept form and standard form of a line?
5. What are other ways to verify the correctness of a line equation?

Tip: When converting between different forms of line equations, be mindful of signs and constants to ensure consistency with standard forms like Ax + By = C.