Through (1,1) and perpendicular to the line 4x − 8y = 1

To find the equation of a line that passes through the point \((1,1)\) and is perpendicular to the line \(4x - 8y = 1\), follow these steps:

### Step 1: Find the slope of the given line
The equation of the line is \(4x - 8y = 1\). We can rewrite this equation in slope-intercept form \(y = mx + c\) to identify the slope \(m\).

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

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

### Step 2: Determine the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. So, the slope \(m_2\) of the perpendicular line is:

\[
m_2 = -\frac{1}{\frac{1}{2}} = -2
\]

### Step 3: Write the equation of the perpendicular line
Now, we use the point-slope form of the equation of a line, which is:

\[
y - y_1 = m(x - x_1)
\]

Substitute the point \((1, 1)\) and the slope \(-2\):

\[
y - 1 = -2(x - 1)
\]

### Step 4: Simplify the equation
Distribute and simplify to put the equation in slope-intercept form:

\[
y - 1 = -2x + 2
\]
\[
y = -2x + 3
\]

### Final Answer
The equation of the line that passes through \((1,1)\) and is perpendicular to \(4x - 8y = 1\) is:

\[
\boxed{y = -2x + 3}
\]

Would you like more details or have any questions about this solution? 

Here are some related questions you might find interesting:

1. How do you find the slope of a line given two points?
2. What is the equation of a line parallel to \(4x - 8y = 1\) passing through \((1, 1)\)?
3. How do you determine if two lines are perpendicular or parallel?
4. Can you derive the equation of a line using different forms, such as point-slope or standard form?
5. What are the properties of lines with negative slopes?

**Tip:** Always remember that the product of the slopes of two perpendicular lines is \(-1\). This is a key concept when determining if two lines are perpendicular.

Discover how to find the equation of a line that passes through the point (1, 1) and is perpendicular to the line 4x - 8y = 1. This problem involves calculating slopes, using the point-slope form of a line equation, and understanding perpendicular relationships between lines.

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