To determine the equations that represent the line perpendicular to the given line \( 5x - 2y = -6 \) and passing through the point \( (5, -4) \), we need to follow these key steps:
1. Find the slope of the given line:
The given line equation is \( 5x - 2y = -6 \). We first rewrite it in slope-intercept form \( y = mx + b \).
[tex]\[
5x - 2y = -6 \implies - 2y = -5x - 6 \implies y = \frac{5}{2}x + 3
\][/tex]
The slope \( m_1 \) of the given line is therefore \( \frac{5}{2} \).
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original slope. So, the slope \( m_2 \) of the line perpendicular to the given line is:
[tex]\[
m_2 = -\frac{1}{(5/2)} = -\frac{2}{5}
\][/tex]
3. Write the equation of the line with slope \( m_2 \) that passes through the point \( (5, -4) \):
We use the point-slope form \( y - y_1 = m(x - x_1) \) to write this equation.
[tex]\[
y - (-4) = -\frac{2}{5}(x - 5)
\][/tex]
Simplifying, we get:
[tex]\[
y + 4 = -\frac{2}{5}(x - 5)
\][/tex]
4. Convert the point-slope form to slope-intercept form (if necessary):
To get another possible form, we can distribute and isolate \( y \):
[tex]\[
y + 4 = -\frac{2}{5}x + 2 \implies y = -\frac{2}{5}x - 2
\][/tex]
5. Convert the equation to standard form (if necessary):
We can multiply the slope-intercept form by 5 to clear the fraction:
[tex]\[
y = -\frac{2}{5}x - 2 \implies 5y = -2x - 10 \implies 2x + 5y = -10
\][/tex]
Given these conversions, the three possible equations of the line perpendicular to \( 5x - 2y = -6 \) and passing through the point \( (5, -4) \) are:
[tex]\[
y = -\frac{2}{5}x - 2,
\][/tex]
[tex]\[
2x + 5y = -10,
\][/tex]
[tex]\[
y + 4 = -\frac{2}{5}(x - 5)
\][/tex]
Therefore, the correct options are:
1. \( y = -\frac{2}{5}x - 2 \)
2. \( 2x + 5y = -10 \)
4. [tex]\( y + 4 = -\frac{2}{5}(x - 5) \)[/tex]
