Certainly! Let's find the equation of the line that is perpendicular to the given line and passes through the point \((-4, 3)\).
### Step-by-Step Solution
1. Identify the given points and find the slope of the given line:
The given points are \((-4, -3)\) and \((4, 1)\).
2. Calculate the slope of the given line:
The slope \(m_1\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
[tex]\[
m_1 = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the coordinates:
[tex]\[
m_1 = \frac{1 - (-3)}{4 - (-4)} = \frac{1 + 3}{4 + 4} = \frac{4}{8} = \frac{1}{2}
\][/tex]
3. Find the slope of the line perpendicular to the given line:
The slope \(m_2\) of the line perpendicular to the given line is the negative reciprocal of \(m_1\):
[tex]\[
m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{1}{2}} = -2
\][/tex]
4. Use the point-slope form to write the equation of the perpendicular line:
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, the point is \((-4, 3)\) and the slope \(m_2 = -2\). Substituting these values into the point-slope form:
[tex]\[
y - 3 = -2(x + 4)
\][/tex]
### Conclusion
The equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point \((-4, 3)\) is:
[tex]\[
y - 3 = -2(x + 4)
\][/tex]
Thus, the correct multiple-choice answer is:
[tex]$[tex]$\boxed{y - 3 = -2(x + 4)}$[/tex]$[/tex]
