Answer:
[tex]y = -\frac{3}{2}x -1[/tex]
Step-by-step explanation:
Given
Perpendicular to [tex]y = \frac{2}{3}x + 2[/tex]
Pass through [tex](-2,2)[/tex]
Required
Determine the line equation
First, we need to determine the slope of [tex]y = \frac{2}{3}x + 2[/tex]
An equation is of the form:
[tex]y = mx + b[/tex]
Where
[tex]m = slope[/tex]
In this case:
[tex]m = \frac{2}{3}[/tex]
Next, we determine the slope of the second line.
Since both lines are perpendicular, the second line has a slope of:
[tex]m_1 = \frac{-1}{m}[/tex]
[tex]m_1 = \frac{-1}{2/3}[/tex]
[tex]m_1 = -\frac{3}{2}[/tex]
Since this line passes through (-2,2); The equation is calculated as thus:
[tex]y - y_1 = m_1(x - x_1)[/tex]
Where
[tex](x_1,y_1) = (-2,2)[/tex]
This gives:
[tex]y - 2 = -\frac{3}{2}(x - (-2))[/tex]
[tex]y - 2 = -\frac{3}{2}(x +2)[/tex]
[tex]y - 2 = -\frac{3}{2}x -3[/tex]
Add 2 to both sides
[tex]y - 2+2 = -\frac{3}{2}x -3 + 2[/tex]
[tex]y = -\frac{3}{2}x -1[/tex]
