To find the equation of the line that passes through the point [tex]\((9, -10)\)[/tex] and is perpendicular to the line with equation [tex]\(y = -\frac{4}{3}x + 14\)[/tex], follow these steps:
1. Identify the slope of the given line:
The slope of the given line [tex]\(y = -\frac{4}{3}x + 14\)[/tex] is [tex]\(-\frac{4}{3}\)[/tex].
2. Find the perpendicular slope:
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line.
So, if the original slope is [tex]\(-\frac{4}{3}\)[/tex], the perpendicular slope ([tex]\(m\)[/tex]) is:
[tex]\[
m = -\frac{1}{-\frac{4}{3}} = \frac{3}{4}
\][/tex]
3. Use the point-slope form of the equation:
The point-slope form of a line's equation 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 the given point.
Plugging in the perpendicular slope [tex]\(m = \frac{3}{4}\)[/tex] and the point [tex]\((9, -10)\)[/tex], we get:
[tex]\[
y - (-10) = \frac{3}{4}(x - 9)
\][/tex]
Simplify:
[tex]\[
y + 10 = \frac{3}{4}(x - 9)
\][/tex]
4. Distribute and simplify:
Now, distribute [tex]\(\frac{3}{4}\)[/tex] to [tex]\(x - 9\)[/tex]:
[tex]\[
y + 10 = \frac{3}{4}x - \frac{3}{4} \cdot 9
\][/tex]
[tex]\[
y + 10 = \frac{3}{4}x - \frac{27}{4}
\][/tex]
Subtract 10 from both sides to isolate [tex]\(y\)[/tex]:
[tex]\[
y = \frac{3}{4}x - \frac{27}{4} - 10
\][/tex]
Express [tex]\(-10\)[/tex] with a common denominator:
[tex]\[
-10 = -\frac{40}{4}
\][/tex]
Substitute this back into the equation:
[tex]\[
y = \frac{3}{4}x - \frac{27}{4} - \frac{40}{4}
\][/tex]
Combine the fractions:
[tex]\[
y = \frac{3}{4}x - \frac{67}{4}
\][/tex]
Thus, the equation of the line that passes through [tex]\((9, -10)\)[/tex] and is perpendicular to [tex]\(y = -\frac{4}{3}x + 14\)[/tex] is:
[tex]\[
y = \frac{3}{4}x - \frac{67}{4}
\][/tex]
