To find the equation of the line through the point (3, 2) and perpendicular to the given line [tex]\(3x + 5y = 15\)[/tex], follow these steps:
### 1. Find the Slope of the Given Line
The equation of the given line is in the form [tex]\(3x + 5y = 15\)[/tex]. To find the slope, we can rewrite this in the slope-intercept form [tex]\(y = mx + b\)[/tex].
[tex]\[
3x + 5y = 15 \implies 5y = -3x + 15 \implies y = -\frac{3}{5}x + 3
\][/tex]
So, the slope ([tex]\(m\)[/tex]) of the given line is [tex]\(-\frac{3}{5}\)[/tex].
### 2. Find 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. The negative reciprocal of [tex]\(-\frac{3}{5}\)[/tex] is [tex]\(\frac{5}{3}\)[/tex].
### 3. Use the Point-Slope Form to Find the Equation of the Perpendicular Line
We have a point [tex]\((3, 2)\)[/tex] and the slope [tex]\(\frac{5}{3}\)[/tex]. The point-slope form of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Plug in the values:
[tex]\[
y - 2 = \frac{5}{3}(x - 3)
\][/tex]
### 4. Convert to Slope-Intercept Form
Now, simplify the equation to get it into the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y - 2 = \frac{5}{3}(x - 3) \\
y - 2 = \frac{5}{3}x - 5 \\
y = \frac{5}{3}x - 5 + 2 \\
y = \frac{5}{3}x - 3
\][/tex]
So, the equation of the line through the point [tex]\((3, 2)\)[/tex] and perpendicular to [tex]\(3x + 5y = 15\)[/tex] is:
[tex]\[
y = \frac{5}{3}x - 3
\][/tex]
In fraction form without spaces, it is written as:
[tex]\[
y=5/3x-3
\][/tex]
