Let's start solving the problem by identifying the pertinent details:
1. Idenfity the slope of the given line [tex]$x + 5y = 7$[/tex]:
- Rewrite the equation in slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
5y = -x + 7 \implies y = -\frac{1}{5}x + \frac{7}{5}
\][/tex]
- The slope [tex]\(m\)[/tex] of this line is [tex]\(-\frac{1}{5}\)[/tex].
2. Find the slope of the line perpendicular to [tex]$x + 5y = 7$[/tex]:
- The slope of the perpendicular line is the negative reciprocal of [tex]\(-\frac{1}{5}\)[/tex], which is [tex]\(5\)[/tex].
3. Use the point-slope form of the line equation:
- Given the point [tex]\((3, -4)\)[/tex] and the slope [tex]\(5\)[/tex], we use the point-slope form:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
- Plug in [tex]\(m = 5\)[/tex], [tex]\((x_1, y_1) = (3, -4)\)[/tex]:
[tex]\[
y - (-4) = 5(x - 3) \implies y + 4 = 5x - 15 \implies y = 5x - 19
\][/tex]
4. Find the value for [tex]\(p\)[/tex]:
- We know that the point [tex]\((2, p)\)[/tex] lies on the line [tex]\(y = 5x - 19\)[/tex].
- Substitute [tex]\(x = 2\)[/tex] into the equation [tex]\(y = 5x - 19\)[/tex]:
[tex]\[
y = 5(2) - 19 \implies y = 10 - 19 \implies y = -9
\][/tex]
- Therefore, [tex]\(p = -9\)[/tex].
So, the value of [tex]\(p\)[/tex] is
[tex]\[
\boxed{-9}
\][/tex]
