To solve this problem, we'll start by finding the slope of the given line that passes through the points [tex]\((-4, -3)\)[/tex] and [tex]\((4, 1)\)[/tex]. The slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated as:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points:
[tex]\[ m = \frac{1 - (-3)}{4 - (-4)} = \frac{1 + 3}{4 + 4} = \frac{4}{8} = \frac{1}{2} \][/tex]
So, the slope [tex]\(m\)[/tex] of the given line is [tex]\(\frac{1}{2}\)[/tex].
Next, we need to find the slope of the line that is perpendicular to this given line. The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. The negative reciprocal of [tex]\(\frac{1}{2}\)[/tex] is:
[tex]\[ m_{\text{perpendicular}} = -\frac{1}{\frac{1}{2}} = -2 \][/tex]
Thus, the slope of the line perpendicular to the original line is [tex]\(-2\)[/tex].
We are also given that this perpendicular line passes through the point [tex]\((-4, 3)\)[/tex]. To find the equation of this perpendicular line in point-slope form, we use the point-slope formula:
[tex]\[ y - y_1 = m_{\text{perpendicular}} (x - x_1) \][/tex]
Here, [tex]\((x_1, y_1) = (-4, 3)\)[/tex] and [tex]\(m_{\text{perpendicular}} = -2\)[/tex]. Substituting these values into the point-slope formula:
[tex]\[ y - 3 = -2 (x + 4) \][/tex]
This matches with one of the provided options. Therefore, the equation of the line that is perpendicular to the given line and passes through the point [tex]\((-4, 3)\)[/tex] is:
[tex]\[ y - 3 = -2 (x + 4) \][/tex]
So, the correct answer is [tex]\(y - 3 = -2 (x + 4)\)[/tex].
