Let's solve this step-by-step.
First, we are given the line equation [tex]\(3x + 5y = -9\)[/tex].
### Step 1: Find the slope of the given line
The general form of the line equation is [tex]\(Ax + By = C\)[/tex], and the slope of this line can be found using [tex]\(-\frac{A}{B}\)[/tex].
For the given line [tex]\(3x + 5y = -9\)[/tex]:
- [tex]\(A = 3\)[/tex]
- [tex]\(B = 5\)[/tex]
Hence, the slope [tex]\(m_1\)[/tex] of the given line is:
[tex]\[ m_1 = -\frac{A}{B} = -\frac{3}{5} = -0.6 \][/tex]
### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope.
Given the slope [tex]\(m_1 = -0.6\)[/tex], the slope [tex]\(m_2\)[/tex] of the perpendicular line will be:
[tex]\[ m_2 = \frac{1}{m_1} = \frac{1}{-0.6} = -\frac{5}{3} \approx -1.67 \][/tex]
### Step 3: Use the point-slope form of the equation of the line
We are given that the perpendicular line passes through the point [tex]\((3, 0)\)[/tex]. The point-slope form of a line is given by:
[tex]\[ y - y_1 = m_2 (x - x_1) \][/tex]
Here, [tex]\( x_1 = 3 \)[/tex], [tex]\( y_1 = 0 \)[/tex], and [tex]\( m_2 = -\frac{5}{3} \)[/tex]. Substituting these values into the point-slope form:
[tex]\[ y - 0 = -\frac{5}{3} (x - 3) \][/tex]
### Step 4: Simplify to slope-intercept form
Simplifying the equation:
[tex]\[ y = -\frac{5}{3} x + \frac{5}{3} \times 3 \][/tex]
[tex]\[ y = -\frac{5}{3} x + 5 \][/tex]
### Step 5: Convert to standard form
To express this in the standard form [tex]\(Ax + By = C\)[/tex]:
[tex]\[ y = -\frac{5}{3} x + 5 \][/tex]
Multiply through by 3 to clear the fraction:
[tex]\[ 3y = -5x + 15 \][/tex]
Rearrange to standard form:
[tex]\[ 5x - 3y = 15 \][/tex]
### Conclusion
The equation of the line that is perpendicular to the given line and passes through the point [tex]\((3,0)\)[/tex] is:
[tex]\[ \boxed{5x - 3y = 15} \][/tex]
So, the correct answer is:
[tex]\[ 5x - 3y = 15 \][/tex]
