To find the [tex]$y$[/tex]-intercept of the line perpendicular to the given line [tex]\( y = -\frac{3}{4}x + 5 \)[/tex] and passing through the point [tex]\((-3, -3)\)[/tex], follow these steps:
1. Determine the slope of the given line:
The slope of the given line [tex]\( y = -\frac{3}{4}x + 5 \)[/tex] is [tex]\( -\frac{3}{4} \)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope.
- The negative reciprocal of [tex]\( -\frac{3}{4} \)[/tex] is [tex]\( \frac{4}{3} \)[/tex].
3. Equation of the perpendicular line:
The equation of a line in point-slope form is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
- Given the point [tex]\((-3, -3)\)[/tex] and the slope [tex]\( \frac{4}{3} \)[/tex], the equation becomes:
[tex]\[ y - (-3) = \frac{4}{3}(x - (-3)) \][/tex]
[tex]\[ y + 3 = \frac{4}{3}(x + 3) \][/tex]
4. Convert to slope-intercept form [tex]\( y = mx + b \)[/tex] to find the [tex]\( y \)[/tex]-intercept:
- Distribute the slope on the right-hand side:
[tex]\[ y + 3 = \frac{4}{3}x + \frac{4}{3} \times 3 \][/tex]
- Simplify inside the parenthesis:
[tex]\[ y + 3 = \frac{4}{3}x + 4 \][/tex]
- Isolate [tex]\( y \)[/tex] by subtracting 3 from both sides:
[tex]\[ y = \frac{4}{3}x + 4 - 3 \][/tex]
[tex]\[ y = \frac{4}{3}x + 1 \][/tex]
5. Identify the [tex]\( y \)[/tex]-intercept:
- The [tex]\( y \)[/tex]-intercept is the constant term [tex]\( b \)[/tex] when the equation is in the form [tex]\( y = mx + b \)[/tex].
- Here, the [tex]\( y \)[/tex]-intercept is 1.
Thus, 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].
So, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
