To determine the equation of the line that passes through the point [tex]\((4,11)\)[/tex] and is perpendicular to the line given by the equation [tex]\(y = \frac{4}{3}x + 7\)[/tex], we can follow these steps:
1. Identify the slope of the given line:
The equation [tex]\(y = \frac{4}{3} x + 7\)[/tex] is in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. Therefore, the slope of the given line is [tex]\(\frac{4}{3}\)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. Hence, we need to find the negative reciprocal of [tex]\(\frac{4}{3}\)[/tex]:
[tex]\[
m_{\perp} = -\frac{1}{m} = -\frac{1}{\frac{4}{3}} = -\frac{3}{4}
\][/tex]
Therefore, the slope of the perpendicular line is [tex]\(-\frac{3}{4}\)[/tex].
3. Use the point-slope form to find the equation of the perpendicular line:
The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\(m\)[/tex] is the slope of the line and [tex]\((x_1, y_1)\)[/tex] is a point on the line. For our perpendicular line, [tex]\(x_1 = 4\)[/tex], [tex]\(y_1 = 11\)[/tex], and [tex]\(m_{\perp} = -\frac{3}{4}\)[/tex]. Substituting these values, we get:
[tex]\[
y - 11 = -\frac{3}{4}(x - 4)
\][/tex]
4. Simplify the equation:
To put it in slope-intercept form ([tex]\(y = mx + b\)[/tex]), we distribute and simplify:
[tex]\[
y - 11 = -\frac{3}{4}x + \frac{3}{4}(4)
\][/tex]
[tex]\[
y - 11 = -\frac{3}{4}x + 3
\][/tex]
Adding 11 to both sides, we get:
[tex]\[
y = -\frac{3}{4}x + 14
\][/tex]
Therefore, the equation of the line that passes through the point [tex]\((4, 11)\)[/tex] and is perpendicular to the line [tex]\(y = \frac{4}{3}x + 7\)[/tex] is [tex]\(y = -\frac{3}{4}x + 14\)[/tex].
The correct answer is:
[tex]\[
\boxed{D. \, y = -\frac{3}{4} x + 14}
\][/tex]
