Answer:
[tex]y_p=\frac{2}{3} x+3[/tex]
Step-by-step explanation:
In the first step, re-write the equation of the given line in slope-y_intercept form in order to see clearly what its slope is. This means to solve for "y" in the equation:
[tex]3x+2y=8\\2y=-3x+8\\y=-\frac{3}{2} x+\frac{8}{2} = -\frac{3}{2} x+4[/tex]
So the slope of the given line is "[tex]-\frac{3}{2}[/tex]"
Recall that the perpendicular line to a given one has a slope that equals the "opposite of the reciprocal" of the original line's slope. This means that the slope of the perpendicular line to our original line must be: "[tex]\frac{2}{3}[/tex]"
We now try to write the equation of the perpendicular line using its slope-y_intercept form, and notice that all we need to find is what is is y_intercept (b):
[tex]y_p=\frac{2}{3} x+b[/tex]
To determine "b" we use the information they give us about this perpendicular line containing the point (0,3):
[tex]y_p=\frac{2}{3} x+b\\3=\frac{2}{3} (0)+b\\3=0+b\\b=3[/tex]
Then we found that b must be 3, and we can now write the complete equation of the perpendicular line:
[tex]y_p=\frac{2}{3} x+3[/tex]
