To find the equation of the line that passes through $(3, 7)$ and is perpendicular to the given line $2x + 4y = 8$, follow these steps:

Step 1: Find the slope of the given line

First, rearrange the given equation $2x + 4y = 8$ into slope-intercept form $y = mx + b$, where $m$ is the slope.

$2x + 4y = 8$ $4y = -2x + 8$ $y = -\frac{1}{2}x + 2$

So, the slope $m_1$ of the given line is $-\frac{1}{2}$.

Step 2: Find the slope of the perpendicular line

The slopes of two perpendicular lines are negative reciprocals of each other. The slope of the line we're looking for will be the negative reciprocal of $-\frac{1}{2}$.

$m_2 = -\frac{1}{\left(-\frac{1}{2}\right)} = 2$

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

Now that we know the slope of the perpendicular line is $2$ and it passes through the point $(3, 7)$, use the point-slope form of a line equation:

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

Substitute $m = 2$, $x_1 = 3$, and $y_1 = 7$:

$y - 7 = 2(x - 3)$

Step 4: Simplify the equation

Expand and simplify the equation:

$y - 7 = 2x - 6$ $y = 2x + 1$

Thus, the equation of the line that passes through $(3, 7)$ and is perpendicular to $2x + 4y = 8$ is:

$y = 2x + 1$

Would you like any further clarification or details?

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Here are five related questions to deepen your understanding:

1. What would the equation of the line be if it were parallel to $2x + 4y = 8$?
2. How do you find the distance between two parallel lines?
3. Can you generalize how to find perpendicular slopes for any linear equation?
4. How does the point-slope form compare to the slope-intercept form for different types of problems?
5. What is the significance of negative reciprocals in perpendicular lines?

Tip: Always simplify the given equation into slope-intercept form first to easily identify the slope when solving line-related problems.