Answer:
y = -2 /3 x - 6
Explanation:
Here we remind ourselves that if we have an equation of the form
[tex]y=mx+b[/tex]
then the equation of a line perpendicular to the above line is
[tex]y=-\frac{1}{m}x+c[/tex]
where c is the y-intercept.
Now for our case, the equation we have is
[tex]3x-2y=5[/tex]
which isn't helpful since we cannot use it to find the equation of the perpendicular line.
Therefore, to make it useful, we first convert it to the slope-intercept form: y = mx + b.
Now, subtracting 3x from both sides gives
[tex]3x-2y-3x=5-3x[/tex][tex]-2y=5-3x[/tex]
dividing both sides by -2 gives
[tex]\frac{-2y}{-2}=\frac{5-3x}{-2}[/tex][tex]y=\frac{3}{2}x-5[/tex]
Now that our equation is in slope-intercept form, we can find the equation for the perpendicular line.
The equation of the line that is prependicular to the above line is
[tex]y=-\frac{1}{3/2}x+b[/tex][tex]y=-\frac{2}{3}x+b[/tex]
Now, we are told that this line must pass through (0, -6). Therefore, we have to find a value of b such that the above line passes through (0, -6). To find b, we put x = 0 and y = -6 into the above equation to get
[tex]-6=-\frac{2}{3}(0)+b[/tex][tex]-6=b\text{.}[/tex]
The value of b is -6.
Therefore, the equation of a line perpendicular to 3x - 2y = 5 line passing through (0, -6) is
[tex]\boxed{y=-\frac{2}{3}x-6.}[/tex]
