Let's work through the problem step by step:

Given Information:

1. Point: $(-5, 3)$
2. Equation of the line: $y = -5x - 4$

Objective:

We need to find the equation of a line that:

• Passes through the point $(-5, 3)$
• Is perpendicular to the given line $y = -5x - 4$

Step 1: Identify the slope of the given line

The equation of the given line is in slope-intercept form, $y = mx + b$, where $m$ is the slope. For the line $y = -5x - 4$, the slope $m = -5$.

Step 2: Find the slope of the perpendicular line

The slopes of perpendicular lines are negative reciprocals of each other. If the slope of one line is $m$, the slope of the line perpendicular to it will be $-\frac{1}{m}$.

Thus, the slope of the line perpendicular to $y = -5x - 4$ is: $m_{\text{perpendicular}} = -\frac{1}{-5} = \frac{1}{5}$

Step 3: Use the point-slope form to find the equation

Now, we know the slope of the perpendicular line is $\frac{1}{5}$, and it passes through the point $(-5, 3)$.

The point-slope form of the equation of a line is: $y - y_1 = m(x - x_1)$ Where $(x_1, y_1)$ is the point and $m$ is the slope.

Substitute $(-5, 3)$ for $(x_1, y_1)$ and $\frac{1}{5}$ for $m$: $y - 3 = \frac{1}{5}(x + 5)$

Step 4: Simplify to slope-intercept form

Expand the equation: $y - 3 = \frac{1}{5}x + 1$

Add 3 to both sides: $y = \frac{1}{5}x + 1 + 3$ $y = \frac{1}{5}x + 4$

Final Answer:

The equation of the line in slope-intercept form is: $y = \frac{1}{5}x + 4$

So, the correct choice is B, and the answer in the box should be: $y = \frac{1}{5}x + 4$

Would you like more details on any of the steps? Here are five related questions to deepen your understanding:

1. What is the general formula for the slope of a line perpendicular to a given slope $m$?
2. How would you solve this if the given line was vertical or horizontal?
3. How can you check if two lines are perpendicular using their slopes?
4. What is the importance of using the point-slope form of a line equation?
5. How do you convert from point-slope form to slope-intercept form in general?

Tip: Always remember that the slopes of perpendicular lines multiply to $-1$. This can be a quick way to check your work!