Problem Analysis:

We are tasked with finding the linear equation that passes through the points $(4, -5)$ and $(12, -9)$. The general equation of a line in slope-intercept form is:

$y = mx + b$

where $m$ is the slope and $b$ is the $y$-intercept.

Step 1: Find the slope $m$

The slope formula is given by:

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

Using the points $(x_1, y_1) = (4, -5)$ and $(x_2, y_2) = (12, -9)$, we compute:

$m = \frac{-9 - (-5)}{12 - 4} = \frac{-9 + 5}{8} = \frac{-4}{8} = -\frac{1}{2}$

Thus, the slope is $m = -\frac{1}{2}$.

Step 2: Use the slope-intercept form to find $b$

The equation of the line is:

$y = mx + b$

Substitute $m = -\frac{1}{2}$ and one of the points, say $(4, -5)$, into the equation to solve for $b$:

$-5 = -\frac{1}{2}(4) + b$

Simplify:

$-5 = -2 + b \quad \Rightarrow \quad b = -5 + 2 = -3$

Thus, the equation of the line is:

$y = -\frac{1}{2}x - 3$

Step 3: Match the equation with the options

The correct answer is:

A. $y = -\frac{1}{2}x - 3$

Final Answer:

A

Would you like me to explain any part in more detail? Let me know!

Related Questions:

1. How do you find the slope of a line between two points geometrically?
2. Can you write the equation of a line given its slope and one point?
3. What is the difference between slope-intercept form and point-slope form?
4. How can we verify that a given equation passes through specific points?
5. How does changing the slope $m$ affect the angle of the line?

Tip:

Always double-check your calculations by plugging the given points back into the final equation to confirm that they satisfy it!