To find the [tex]\( y \)[/tex]-intercept of the line perpendicular to the line [tex]\( y = -\frac{3}{4} x + 5 \)[/tex] and passing through the point [tex]\((-3, -3)\)[/tex], we can follow these steps:
1. Find the slope of the given line:
The given line has the equation [tex]\( y = -\frac{3}{4} x + 5 \)[/tex].
- The slope ([tex]\( m \)[/tex]) of this line is [tex]\( -\frac{3}{4} \)[/tex].
2. Determine the slope of the perpendicular line:
- The slopes of perpendicular lines are negative reciprocals of each other.
- Therefore, the slope of the perpendicular line is the negative reciprocal of [tex]\( -\frac{3}{4} \)[/tex]. The negative reciprocal of [tex]\( -\frac{3}{4} \)[/tex] is [tex]\( \frac{4}{3} \)[/tex].
3. Use the point-slope form to find the equation of the perpendicular line:
- The point-slope form of a line is [tex]\( y - y_1 = m (x - x_1) \)[/tex].
- We have the point [tex]\((-3, -3)\)[/tex] and the slope [tex]\( \frac{4}{3} \)[/tex].
Plugging in these values, we get:
[tex]\[
y - (-3) = \frac{4}{3} (x - (-3))
\][/tex]
Simplifying, we have:
[tex]\[
y + 3 = \frac{4}{3} (x + 3)
\][/tex]
4. Solve for the [tex]\( y \)[/tex]-intercept:
- To find the [tex]\( y \)[/tex]-intercept, we need to solve for [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
Using our equation [tex]\( y + 3 = \frac{4}{3} (x + 3) \)[/tex], set [tex]\( x = 0 \)[/tex]:
[tex]\[
y + 3 = \frac{4}{3} (0 + 3)
\][/tex]
[tex]\[
y + 3 = \frac{4}{3} \cdot 3
\][/tex]
[tex]\[
y + 3 = 4
\][/tex]
[tex]\[
y = 4 - 3
\][/tex]
[tex]\[
y = 1
\][/tex]
So, the [tex]\( y \)[/tex]-intercept of the line perpendicular to [tex]\( y = -\frac{3}{4} x + 5 \)[/tex] and passing through the point [tex]\((-3, -3)\)[/tex] is [tex]\( 1 \)[/tex], which corresponds to the option [tex]\(\boxed{1}\)[/tex].
