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The problem asks for the equation of a line that passes through the origin and is perpendicular to the line \( 8x + y = 4 \). Let's solve this step-by-step:

### Step 1: Find the slope of the given line \( 8x + y = 4 \)
To find the slope, we first rewrite the equation in slope-intercept form \( y = mx + b \), where \( m \) is the slope.

\[
8x + y = 4 \quad \text{(Subtract \( 8x \) from both sides)}
\]
\[
y = -8x + 4
\]
So, the slope \( m_1 \) of the given line is \( -8 \).

### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of \( -8 \) is:

\[
m_2 = \frac{1}{8}
\]

### Step 3: Use the point-slope form to find the equation of the new line
Since the new line passes through the origin (0, 0), we can use the slope-intercept form \( y = mx + b \) with \( m = \frac{1}{8} \) and \( b = 0 \) (since it passes through the origin).

\[
y = \frac{1}{8}x
\]

Thus, the equation of the line is:

\[
y = \frac{1}{8}x
\]

### Final Answer:
\[
y = \frac{1}{8}x
\]

Would you like further details on any step, or do you have any questions?

Here are 5 related questions to expand on this topic:
1. What is the geometric significance of the negative reciprocal when finding perpendicular slopes?
2. How do we determine the y-intercept for lines that do not pass through the origin?
3. What would the equation look like if the new line passed through a different point, such as (2, 3)?
4. How can you derive the perpendicular slope using vector dot products?
5. How can this method be extended to three-dimensional coordinate geometry?

**Tip:** Always rewrite equations in slope-intercept form when identifying slopes to ensure clarity.

Find the equation of the line that passes through the origin, perpendicular to 8x + y = 4. Write the equation in slope-intercept form.

Learn how to find the equation of a line that passes through the origin and is perpendicular to 8x + y = 4. This problem covers key algebraic concepts, including the slope-intercept form and perpendicular slopes. Suitable for students in Grades 9-11.

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