Answer:
[tex]y=-\frac{3}{2}x-6[/tex]
Step-by-step explanation:
Since the line needs to be perpendicular to [tex]2x-3y=6[/tex], that means the slope of the line must be the opposite reciprocal. Rearrange the equation [tex]2x-3y=6[/tex] to solve for the value of y, with variable y on the left side.
[tex]-3y=-2x+6\\y=\frac{2}{3} x-2[/tex]
So, the slope of the line given already is [tex]\frac{2}{3}[/tex]. The opposite reciprocal of this is [tex]-\frac{3}{2}[/tex].
From what information we know so far (the slope) about the equation of the line we are trying to find, we can write a basic equation that allows us to solve for the y-intercept. Use the equation [tex]y=mx+b[/tex], where m is the slope (which we already found) and b is the y-intercept.
[tex]y=-\frac{3}{2}x+b[/tex]
Since we are given a set of coordinate points that the line must pass through, we can substitute (-2, -3) in for x and y in the equation above. Then, solve for the value of b, which is our y-intercept.
[tex]-3=-\frac{3}{2}(-2)+b\\ -3=3+b\\b=-6[/tex]
Now we have all the necessary information to create our equation.
[tex]y=-\frac{3}{2}x-6[/tex]
