What is an equation of the line that passes through the point left parenthesis, minus, 8, comma, minus, 6, right parenthesis(−8,−6) and is perpendicular to the line 4, x, minus, y, equals, 64*x*−*y*=6?

To find the equation of the line that passes through the point \((-8, -6)\) and is perpendicular to the line given by \(4x - y = 6\), follow these steps:

### Step 1: Determine the slope of the given line
The given line is:
\[
4x - y = 6
\]
First, rewrite this equation in slope-intercept form (\(y = mx + b\)), where \(m\) is the slope:
\[
y = 4x - 6
\]
So, the slope (\(m\)) of the given line is \(4\).

### Step 2: Find the slope of the perpendicular line
The slope of a line that is perpendicular to another is the negative reciprocal of the slope of the original line. Therefore, the slope of the line perpendicular to the given line is:
\[
m_{\text{perpendicular}} = -\frac{1}{4}
\]

### Step 3: Use the point-slope form of the equation of a line
The point-slope form of the equation of a line is:
\[
y - y_1 = m(x - x_1)
\]
Here, \(m = -\frac{1}{4}\) and the point \((x_1, y_1) = (-8, -6)\). Plugging these into the point-slope form:
\[
y - (-6) = -\frac{1}{4}(x - (-8))
\]
Simplifying this equation:
\[
y + 6 = -\frac{1}{4}(x + 8)
\]
\[
y + 6 = -\frac{1}{4}x - 2
\]
Finally, subtract 6 from both sides to get the equation in slope-intercept form:
\[
y = -\frac{1}{4}x - 8
\]

### Final Answer:
The equation of the line that passes through \((-8, -6)\) and is perpendicular to the line \(4x - y = 6\) is:
\[
y = -\frac{1}{4}x - 8
\]

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

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Here are 5 related questions:
1. How do you find the equation of a line parallel to a given line?
2. What is the importance of the slope-intercept form of a line?
3. How can you determine if two lines are perpendicular just by looking at their slopes?
4. What happens to the equation if the point provided lies on the x-axis?
5. How can you convert the slope-intercept form back to the standard form?

**Tip:** When determining the slope of a line from its equation, always try to rewrite the equation in slope-intercept form. This makes it easier to identify the slope.

Learn how to find the equation of a line that passes through the point (-8, -6) and is perpendicular to the line given by 4x - y = 6. This problem involves understanding slopes, perpendicular lines, and using the point-slope form to derive the equation. Suitable for high school students.

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