To find the equation of a line with a given slope and point, we use the point-slope form of a line equation:

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

where:

• $m = \frac{1}{2}$ (the slope),
• $(x_1, y_1) = (-4, -9)$ (the point the line passes through).

Step 1: Substitute the values

$y - (-9) = \frac{1}{2}(x - (-4))$

which simplifies to:

$y + 9 = \frac{1}{2}(x + 4)$

Step 2: Distribute the slope

$y + 9 = \frac{1}{2}x + \frac{1}{2} \times 4$

$y + 9 = \frac{1}{2}x + 2$

Step 3: Solve for $y$ by isolating it on one side

$y = \frac{1}{2}x + 2 - 9$

$y = \frac{1}{2}x - 7$

Answer

The equation of the line is:

$y = \frac{1}{2}x - 7$

So, the correct option is:

$\text{Option C: } y = \frac{1}{2}x - 7$

Would you like a detailed breakdown of each step or have any other questions?

Here are some related questions to deepen your understanding:

1. How do you find the y-intercept of a line given a point and the slope?
2. What is the general form of a linear equation and how does it relate to the point-slope form?
3. How would the equation change if the slope were negative, say $-\frac{1}{2}$?
4. How can we verify if a given point lies on a specific line?
5. What is the difference between point-slope form and slope-intercept form?

Tip: To quickly verify your solution, substitute the given point into your final equation to see if it satisfies the equation.