To determine the [tex]\(y\)[/tex]-intercept of the line perpendicular to the line [tex]\(y = -\frac{3}{4}x + 5\)[/tex] that passes through the point [tex]\((-3, -3)\)[/tex], follow these steps:
1. Identify the slope of the given line: The given line is [tex]\(y = -\frac{3}{4}x + 5\)[/tex], so the slope (m) is [tex]\(-\frac{3}{4}\)[/tex].
2. Find the slope of the perpendicular line: The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line. Therefore, the slope of the perpendicular line will be:
[tex]\[
\frac{4}{3}
\][/tex]
3. Use the point-slope form of the equation of a line: The point-slope form of a line's equation is [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. Here, the slope [tex]\(m\)[/tex] is [tex]\(\frac{4}{3}\)[/tex] and the point given is [tex]\((-3, -3)\)[/tex].
4. Substitute the given point and slope into the point-slope form equation:
[tex]\[
y - (-3) = \frac{4}{3}(x - (-3))
\][/tex]
This simplifies to:
[tex]\[
y + 3 = \frac{4}{3}(x + 3)
\][/tex]
5. Isolate [tex]\(y\)[/tex] to get the equation of the line in slope-intercept form:
[tex]\[
y + 3 = \frac{4}{3}x + 4
\][/tex]
Now, subtract 3 from both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{4}{3}x + 1
\][/tex]
6. Identify the [tex]\(y\)[/tex]-intercept: In the equation [tex]\(y = \frac{4}{3}x + 1\)[/tex], the [tex]\(y\)[/tex]-intercept is the constant term independent of [tex]\(x\)[/tex]. Therefore, the [tex]\(y\)[/tex]-intercept is [tex]\(1\)[/tex].
Thus, the [tex]\(y\)[/tex]-intercept of the line perpendicular to [tex]\(y = -\frac{3}{4}x + 5\)[/tex] that passes through the point [tex]\((-3, -3)\)[/tex] is [tex]\(1\)[/tex].
