The given line passes through the points  and (4, 1).

On a coordinate plane, a line goes through (negative 4, negative 3) and (4, 1).

What is the equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point (, 3)?

To find the equation of the line that is perpendicular to the given line and passes through a specific point, we can follow these steps:

### 1. Find the Slope of the Given Line
The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

For the given points \((-4, -3)\) and \((4, 1)\):

\[
m = \frac{1 - (-3)}{4 - (-4)} = \frac{1 + 3}{4 + 4} = \frac{4}{8} = \frac{1}{2}
\]

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

### 2. Find 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. Therefore, the slope \( m' \) of the perpendicular line is:

\[
m' = -\frac{1}{m} = -\frac{1}{\frac{1}{2}} = -2
\]

### 3. Write the Equation in Point-Slope Form
The point-slope form of a line's equation is:

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

You mentioned the point the perpendicular line passes through as \((x_1, 3)\). However, it seems like the x-coordinate was not provided. If we assume the point to be \((x_1, 3)\), the equation in point-slope form would be:

\[
y - 3 = -2(x - x_1)
\]

### Conclusion
The equation of the line perpendicular to the given line and passing through the point \((x_1, 3)\) in point-slope form is:

\[
\boxed{y - 3 = -2(x - x_1)}
\]

If you provide the x-coordinate for the point, I can give you the exact equation.

Would you like more details on this process, or do you have any questions? Here are some related questions:

1. How do you find the equation of a line given two points?
2. What is the relationship between the slopes of parallel and perpendicular lines?
3. How can you convert from point-slope form to slope-intercept form?
4. How do you determine the slope of a line from its equation?
5. What is the general formula for the distance between two points on a coordinate plane?

**Tip:** Always remember, for perpendicular lines, the product of their slopes equals \(-1\).

Learn how to find the equation of a line that is perpendicular to another line and passes through a specified point. This problem involves calculating slopes, applying the point-slope form of a line equation, and understanding the relationship between perpendicular slopes. Suitable for students in Grades 7-9.

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