What is the [tex]$y$[/tex]-intercept of the line perpendicular to the line [tex][tex]$y=-\frac{3}{4} x+5$[/tex][/tex] that includes the point [tex][tex]$(-3, -3)$[/tex][/tex]?

A. [tex]-\frac{3}{4}[/tex]
B. 1
C. 7
D. [tex]-\frac{21}{4}[/tex]



Answer :

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].