What is an equation of the line that passes through the point (6,-4) and is perpendicular to the line 2x-y+4?

To find the equation of the line that passes through the point \((6, -4)\) and is perpendicular to the line given by \(2x - y + 4 = 0\), follow these steps:

### Step 1: Determine the slope of the given line
The equation of the given line is \(2x - y + 4 = 0\). First, let's rewrite it in slope-intercept form (\(y = mx + b\)):

\[
2x - y + 4 = 0 \quad \text{(subtract 2x from both sides)}
\]
\[
-y = -2x - 4 \quad \text{(multiply through by -1)}
\]
\[
y = 2x + 4
\]

The slope \(m_1\) of the given line is \(2\).

### 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. So, if the slope of the given line is \(m_1 = 2\), the slope \(m_2\) of the perpendicular line will be:

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

### Step 3: Use the point-slope form to find the equation of the perpendicular line
We now have the slope \(m_2 = -\frac{1}{2}\) and the point \((6, -4)\). The point-slope form of the equation of a line is:

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

Substitute \(m = -\frac{1}{2}\), \(x_1 = 6\), and \(y_1 = -4\):

\[
y - (-4) = -\frac{1}{2}(x - 6)
\]
\[
y + 4 = -\frac{1}{2}(x - 6)
\]

### Step 4: Simplify the equation
Distribute the slope on the right side:

\[
y + 4 = -\frac{1}{2}x + 3
\]

Subtract 4 from both sides to solve for \(y\):

\[
y = -\frac{1}{2}x + 3 - 4
\]
\[
y = -\frac{1}{2}x - 1
\]

### Final Answer
The equation of the line that passes through the point \((6, -4)\) and is perpendicular to the line \(2x - y + 4 = 0\) is:

\[
\boxed{y = -\frac{1}{2}x - 1}
\]

Would you like any details or further explanations?

### Related Questions:
1. How do you find the equation of a line given two points?
2. What is the general form of the equation of a line?
3. How do you determine if two lines are parallel?
4. What is the importance of the slope-intercept form of a line?
5. How do you convert a linear equation from standard form to slope-intercept form?
6. What are the characteristics of perpendicular lines in a coordinate plane?
7. How do you find the distance between a point and a line?
8. Can you derive the point-slope form from the slope-intercept form?

**Tip:** When working with perpendicular lines, always remember to use the negative reciprocal of the slope of the original line to find the slope of the perpendicular line.

Learn how to find the equation of a line perpendicular to 2x - y + 4 passing through the point (6, -4). This problem involves using the point-slope form and understanding perpendicular slopes. Suitable for students in Grades 9-12.

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