To determine the equation of a line perpendicular to the given line [tex]\(y = \frac{4}{3} x - 16\)[/tex] and passing through the point [tex]\((1, -9)\)[/tex], follow these steps:
1. Find the slope of the given line:
The given equation is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] represents the slope. So, the slope of the given line is:
[tex]\[
\text{slope} = \frac{4}{3}
\][/tex]
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original slope:
[tex]\[
\text{slope of perpendicular line} = -\frac{1}{\left(\frac{4}{3}\right)} = -\frac{3}{4}
\][/tex]
3. Use the point-slope form to find the equation of the perpendicular line:
The point-slope form of a line's equation is:
[tex]\[
y - y_1 = m (x - x_1)
\][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] is the given point [tex]\((1, -9)\)[/tex] and [tex]\(m\)[/tex] is the slope of the perpendicular line [tex]\(-\frac{3}{4}\)[/tex]. Plugging in these values:
[tex]\[
y - (-9) = -\frac{3}{4}(x - 1)
\][/tex]
Simplifying:
[tex]\[
y + 9 = -\frac{3}{4}x + \frac{3}{4}
\][/tex]
4. Convert the equation to slope-intercept form:
Subtract 9 from both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{3}{4}x + \frac{3}{4} - 9
\][/tex]
Simplify the constant term on the right-hand side:
[tex]\[
y = -\frac{3}{4}x + \frac{3}{4} - \frac{36}{4}
\][/tex]
[tex]\[
y = -\frac{3}{4}x - \frac{33}{4}
\][/tex]
5. Convert the equation to standard form [tex]\(Ax + By = C\)[/tex]:
To eliminate the fractions, multiply every term by 4:
[tex]\[
4y = -3x - 33
\][/tex]
Rearrange to bring all terms to one side:
[tex]\[
3x + 4y = -33
\][/tex]
Thus, the correct equation of the line in standard form, perpendicular to the line [tex]\(y = \frac{4}{3}x - 16\)[/tex] and passing through the point [tex]\((1, -9)\)[/tex], is:
[tex]\[
3x + 4y = -33
\][/tex]
