Answer:
The line equation in the slope-intercept form:
![y=\frac{3}{2}x+2](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B3%7D%7B2%7Dx%2B2)
Step-by-step explanation:
We know that the slope-intercept of line equation is
![y = mx+b](https://tex.z-dn.net/?f=y%20%3D%20mx%2Bb)
Where m is the slope and b is the y-intercept
Given the line
![2x + 3y = 9](https://tex.z-dn.net/?f=2x%20%2B%203y%20%3D%209)
Writing in the slope-intercept form
![2x + 3y = 9](https://tex.z-dn.net/?f=2x%20%2B%203y%20%3D%209)
![y=-\frac{2}{3}x+3](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B2%7D%7B3%7Dx%2B3)
Therefore, the slope of the line = m = -2/3
We know that a line perpendicular to another line contains a slope that is the negative reciprocal of the slope of the other line, such as:
slope = m = -2/3
perpendicular slope = – 1/m
![=-\frac{1}{-\frac{2}{3}}=\frac{3}{2}](https://tex.z-dn.net/?f=%3D-%5Cfrac%7B1%7D%7B-%5Cfrac%7B2%7D%7B3%7D%7D%3D%5Cfrac%7B3%7D%7B2%7D)
Given the point
(x₁, y₁) = (-2, -1)
Using the point-slope form of the line equation
![y-y_1=m\left(x-x_1\right)](https://tex.z-dn.net/?f=y-y_1%3Dm%5Cleft%28x-x_1%5Cright%29)
where m is the slope and (x₁, y₁) is the point
substituting the perpendicular slope m = 3/2 and the point (-2, -1)
![y-\left(-1\right)=\frac{3}{2}\left(x-\left(-2\right)\right)](https://tex.z-dn.net/?f=y-%5Cleft%28-1%5Cright%29%3D%5Cfrac%7B3%7D%7B2%7D%5Cleft%28x-%5Cleft%28-2%5Cright%29%5Cright%29)
Writing in the slope-intercept form
![y+1=\frac{3}{2}\left(x+2\right)](https://tex.z-dn.net/?f=y%2B1%3D%5Cfrac%7B3%7D%7B2%7D%5Cleft%28x%2B2%5Cright%29)
subtract 1 from both sides
![y+1-1=\frac{3}{2}\left(x+2\right)-1](https://tex.z-dn.net/?f=y%2B1-1%3D%5Cfrac%7B3%7D%7B2%7D%5Cleft%28x%2B2%5Cright%29-1)
![y=\frac{3}{2}x+2](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B3%7D%7B2%7Dx%2B2)
Thus, the line equation in the slope-intercept form:
![y=\frac{3}{2}x+2](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B3%7D%7B2%7Dx%2B2)