To find the equation of the line that passes through the point [tex]\((4, 11)\)[/tex] and is perpendicular to the line with the equation [tex]\( y = \frac{4}{3} x + 7 \)[/tex], let's work through the steps for determining the correct equation.
### Step-by-Step Solution:
#### Step 1: Identify the Slope of the Given Line
The given line is [tex]\( y = \frac{4}{3} x + 7 \)[/tex]. The slope of this line is [tex]\( \frac{4}{3} \)[/tex].
#### Step 2: Determine the Slope of the Perpendicular Line
The slope of a line perpendicular to another is the negative reciprocal of the given line's slope. Therefore, we calculate the negative reciprocal of [tex]\( \frac{4}{3} \)[/tex]:
[tex]\[
\text{Slope of perpendicular line} = -\frac{1}{\left(\frac{4}{3}\right)} = -\frac{3}{4}
\][/tex]
#### Step 3: Use the Point-Slope Form of the Equation of a 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 and [tex]\((x_1, y_1)\)[/tex] is a point on the line. We know the slope [tex]\( m = -\frac{3}{4} \)[/tex] and the point [tex]\((x_1, y_1) = (4, 11)\)[/tex].
#### Step 4: Plug in the Known Values
Substituting the slope and point into the point-slope form:
[tex]\[
y - 11 = -\frac{3}{4} (x - 4)
\][/tex]
#### Step 5: Simplify to Slope-Intercept Form
Now, we simplify the equation to get it into the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y - 11 = -\frac{3}{4}x + 3
\][/tex]
Adding 11 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{3}{4}x + 3 + 11
\][/tex]
[tex]\[
y = -\frac{3}{4}x + 14
\][/tex]
#### Step 6: Select the Corresponding Answer
The equation of the line that passes through the point (4,11) and is perpendicular to the line [tex]\( y = \frac{4}{3} x + 7 \)[/tex] is:
[tex]\[
y = -\frac{3}{4} x + 14
\][/tex]
Thus, the correct answer is:
A. [tex]\( y = -\frac{3}{4} x + 14 \)[/tex]
