Sure! Let's find the equation of the line through the point [tex]\((-5, 5)\)[/tex] that is perpendicular to the line given by [tex]\( y = \frac{5}{9} x - 4 \)[/tex].
1. Identify the slope of the given line:
The equation of the given line is [tex]\( y = \frac{5}{9} x - 4 \)[/tex]. The slope of this line is [tex]\(\frac{5}{9}\)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. So, the slope of the line we are trying to find is [tex]\( -\frac{1}{(\frac{5}{9})} \)[/tex].
Simplifying this, we get:
[tex]\[
-\frac{1}{\frac{5}{9}} = -\frac{9}{5}
\][/tex]
3. Use the point-slope form of the equation of a line:
The point-slope form 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.
Here, the slope [tex]\( m \)[/tex] is [tex]\(-\frac{9}{5}\)[/tex] and the point [tex]\((x_1, y_1)\)[/tex] is [tex]\((-5, 5)\)[/tex].
4. Substitute the known values into the point-slope form:
[tex]\[
y - 5 = -\frac{9}{5}(x + 5)
\][/tex]
5. Convert to slope-intercept form (y = mx + b):
Distribute the slope on the right-hand side:
[tex]\[
y - 5 = -\frac{9}{5} x - \frac{9}{5} \cdot 5
\][/tex]
Simplify the right-hand side:
[tex]\[
y - 5 = -\frac{9}{5} x - 9
\][/tex]
6. Isolate [tex]\(y\)[/tex]:
Add 5 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = -\frac{9}{5} x - 9 + 5
\][/tex]
Simplify the constant terms:
[tex]\[
y = -\frac{9}{5} x - 4
\][/tex]
Therefore, the equation of the line that is perpendicular to [tex]\(y = \frac{5}{9} x - 4\)[/tex] and passes through the point [tex]\((-5, 5)\)[/tex] is:
[tex]\[
y = -\frac{9}{5} x - 4
\][/tex]
Simplifying the fraction in numerical form, we get:
[tex]\[
y = -1.7999999999999998 x - 4
\][/tex]
So, the final answer is:
[tex]\[
y = -1.8x - 4
\][/tex]
