To find the equation of the straight line passing through the point [tex]\((0, -1)\)[/tex] which is perpendicular to the line [tex]\( y = -\frac{3}{4}x - 3 \)[/tex], follow these steps:
1. Identify the slope of the given line:
The given line is [tex]\( y = -\frac{3}{4}x - 3 \)[/tex]. Here, the slope of this line is [tex]\(-\frac{3}{4}\)[/tex].
2. Calculate 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:
[tex]\[
\text{slope of the perpendicular line} = -\frac{1}{\left(-\frac{3}{4}\right)} = \frac{4}{3}
\][/tex]
3. Use the point-slope form to determine the equation of the line:
The form of the equation of a line is given by [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Given the point [tex]\((0, -1)\)[/tex] and the slope [tex]\(\frac{4}{3}\)[/tex], we will determine the y-intercept [tex]\( b \)[/tex].
4. Find the y-intercept [tex]\( b \)[/tex]:
Substitute the point [tex]\((0, -1)\)[/tex] into the equation [tex]\( y = \frac{4}{3}x + b \)[/tex]:
[tex]\[
-1 = \frac{4}{3}(0) + b
\][/tex]
[tex]\[
b = -1
\][/tex]
5. Write the final equation of the line:
Now we have the slope [tex]\( \frac{4}{3} \)[/tex] and the y-intercept [tex]\( -1 \)[/tex]. Hence, the equation of the line is:
[tex]\[
y = \frac{4}{3}x - 1
\][/tex]
Therefore, the equation of the straight line passing through [tex]\((0, -1)\)[/tex] and perpendicular to the line [tex]\( y = -\frac{3}{4}x - 3 \)[/tex] is:
[tex]\[
y = \frac{4}{3}x - 1
\][/tex]
