Let's solve the problem step by step.

The goal is to find the equation of the line in slope-intercept form, $y = mx + b$, using the two points $(-1, 7)$ and $(3, -1)$.

Step 1: Find the slope ($m$)

The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

Substitute the given points $(-1, 7)$ and $(3, -1)$:

$m = \frac{-1 - 7}{3 - (-1)} = \frac{-8}{3 + 1} = \frac{-8}{4} = -2$

So, the slope is $m = -2$.

Step 2: Find the $y$-intercept ($b$)

Using the slope-intercept form equation, $y = mx + b$, we can substitute one of the points (e.g., $(-1, 7)$) and the slope $m = -2$ to solve for $b$:

$y = mx + b$

Substitute $y = 7$, $x = -1$, and $m = -2$:

$7 = -2(-1) + b$

$7 = 2 + b$

$b = 7 - 2 = 5$

So, the $y$-intercept is $b = 5$.

Step 3: Write the equation

Now that we know $m = -2$ and $b = 5$, the equation of the line is:

$y = -2x + 5$

Final Answer:

$y = -2x + 5$

Would you like further explanation or details? Here are 5 related questions to expand your understanding:

1. How can you verify if the equation $y = -2x + 5$ passes through both points?
2. What is the general formula for slope-intercept form, and how does it relate to the point-slope formula?
3. What happens to the equation if the slope is $0$ or undefined?
4. How would the equation look in standard form instead of slope-intercept form?
5. What is the significance of the $y$-intercept in a real-world context?

Tip: Always double-check your slope calculation by ensuring that the denominator (change in $x$) is not zero to avoid undefined slopes.