To determine which equations represent a line perpendicular to the line [tex]\(5x - 2y = -6\)[/tex] and passing through the point [tex]\((5, -4)\)[/tex], we need to follow these steps:
1. Find the slope of the original line [tex]\(5x - 2y = -6\)[/tex].
Rearrange the equation to the 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 of the original line is [tex]\(\frac{5}{2}\)[/tex].
2. Determine the slope of the perpendicular line.
The slope of the line perpendicular to a line with slope [tex]\(m\)[/tex] is the negative reciprocal, so if the original slope is [tex]\(\frac{5}{2}\)[/tex], the perpendicular slope is:
[tex]\[
-\frac{1}{\left( \frac{5}{2} \right)} = -\frac{2}{5}
\][/tex]
3. Write the equation of the perpendicular line that passes through point [tex]\((5, -4)\)[/tex] using the point-slope form of a linear equation:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\(m = -\frac{2}{5}\)[/tex], [tex]\( (x_1, y_1) = (5, -4) \)[/tex]:
[tex]\[
y - (-4) = -\frac{2}{5}(x - 5) \implies y + 4 = -\frac{2}{5}(x - 5)
\][/tex]
4. Convert this equation to different forms for comparison:
- Slope-intercept form (express [tex]\(y\)[/tex] as a function of [tex]\(x\)[/tex]):
[tex]\[
y + 4 = -\frac{2}{5}(x - 5) \implies y + 4 = -\frac{2}{5}x + 2 \implies y = -\frac{2}{5}x - 2
\][/tex]
- General form (collect terms on one side):
[tex]\[
y + 4 = -\frac{2}{5}x + 2 \implies 5y + 20 = -2x + 10 \implies 2x + 5y = -10
\][/tex]
5. Compare these forms with the given options:
1. [tex]\(y = -\frac{2}{5}x - 2 \)[/tex]
2. [tex]\(2x + 5y = -10\)[/tex]
3. [tex]\(2x - 5y = -10\)[/tex]
4. [tex]\(y + 4 = -\frac{2}{5}(x - 5)\)[/tex]
5. [tex]\(y - 4 = \frac{5}{2}(x + 5)\)[/tex]
- Option 1: [tex]\(y = -\frac{2}{5}x - 2\)[/tex] matches the slope-intercept form.
- Option 2: [tex]\(2x + 5y = -10\)[/tex] matches the general form.
- Option 3: [tex]\(2x - 5y = -10\)[/tex] does not match.
- Option 4: [tex]\(y + 4 = -\frac{2}{5}(x - 5)\)[/tex] matches the point-slope form.
- Option 5: [tex]\(y - 4 = \frac{5}{2}(x + 5)\)[/tex] does not describe a perpendicular line.
Therefore, the three options that represent the line perpendicular to [tex]\(5x - 2y = -6\)[/tex] and passing through the point [tex]\((5, -4)\)[/tex] are:
[tex]\[
\boxed{ y = -\frac{2}{5}x - 2, \; 2x + 5y = -10, \; y + 4 = -\frac{2}{5}(x - 5)}
\][/tex]
