To determine the equations of the line perpendicular to [tex]\(5x - 2y = -6\)[/tex] that passes through the point [tex]\((5, -4)\)[/tex], let's perform the following steps:
1. Find the slope of the original line:
The given line is [tex]\(5x - 2y = -6\)[/tex]. To find its slope, rewrite it in slope-intercept form [tex]\(y = mx + b\)[/tex].
[tex]\[
5x - 2y = -6 \implies -2y = -5x - 6 \implies y = \frac{5}{2}x + 3
\][/tex]
The slope ([tex]\(m_1\)[/tex]) of the line is [tex]\(\frac{5}{2}\)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Therefore, if the slope of the original line is [tex]\(\frac{5}{2}\)[/tex], the slope ([tex]\(m_2\)[/tex]) of the perpendicular line will be:
[tex]\[
m_2 = -\frac{1}{m_1} = -\frac{1}{\frac{5}{2}} = -\frac{2}{5}
\][/tex]
3. Find the equation of the perpendicular line passing through the point [tex]\((5, -4)\)[/tex]:
Use the point-slope form of the equation of a line [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.
[tex]\[
y - (-4) = -\frac{2}{5}(x - 5)
\][/tex]
Simplify this to get the equation in slope-intercept form:
[tex]\[
y + 4 = -\frac{2}{5}(x - 5)
\][/tex]
[tex]\[
y + 4 = -\frac{2}{5}x + 2
\][/tex]
[tex]\[
y = -\frac{2}{5}x - 2
\][/tex]
So, the equation in slope-intercept form is [tex]\(y = -\frac{2}{5}x - 2\)[/tex].
4. Verify the equation forms:
Let's check the given options:
- Option 1: [tex]\(y = -\frac{2}{5}x - 2\)[/tex] — [tex]\(y = -\frac{2}{5}x - 2\)[/tex]. This matches the equation we found.
- Option 2: [tex]\(2x + 5y = -10\)[/tex]. Convert this to slope-intercept form:
[tex]\[
5y = -2x - 10
\][/tex]
[tex]\[
y = -\frac{2}{5}x - 2
\][/tex]
This also matches the equation.
- Option 3: [tex]\(2x - 5y = -10\)[/tex]. Convert this to slope-intercept form:
[tex]\[
5y = 2x + 10
\][/tex]
[tex]\[
y = \frac{2}{5}x + 2
\][/tex]
This does not match the equation.
- Option 4: [tex]\(y + 4 = -\frac{2}{5}(x - 5)\)[/tex]. This is already in point-slope form and directly simplifies as we showed earlier. It matches the equation.
- Option 5: [tex]\(y - 4 = \frac{5}{2}(x + 5)\)[/tex]. Convert this to slope-intercept form:
[tex]\[
y - 4 = \frac{5}{2}x + \frac{25}{2}
\][/tex]
[tex]\[
y = \frac{5}{2}x + \frac{25}{2} + 4
\][/tex]
This does not match the equation.
Therefore, the correct equations that represent the line that is perpendicular to [tex]\(5x - 2y = -6\)[/tex] and passes through the point [tex]\((5, -4)\)[/tex] are:
1. [tex]\(y = -\frac{2}{5}x - 2\)[/tex]
2. [tex]\(2x + 5y = -10\)[/tex]
4. [tex]\(y + 4 = -\frac{2}{5}(x - 5)\)[/tex]
Hence, the options are [tex]\(\boxed{2}\)[/tex] and [tex]\(\boxed{4}\)[/tex].
