Consider the line $y=\frac{2}{9} x-6$. Find the equation of the line that is perpendicular to this line and passes through the point $(4,6)$. Find the equation of the line that is parallel to this line and passes through the point $(4,6)$. Note that the ALEKS graphing calculator may be helpful in checking your answer. Equation of perpendicular line: $\square$ \[ \text { 믐 } \quad \square=\square \] Equation of parallel line: \[ \times 5 \]

Step 1 ：The slope of the given line is \( \frac{2}{9} \).

Step 2 ：The slope of a line perpendicular to this would be the negative reciprocal, which is \( -\frac{9}{2} \).

Step 3 ：The slope of a line parallel to this would be the same as the original line, which is \( \frac{2}{9} \).

Step 4 ：We can use the point-slope form of a line, \( y - y_1 = m(x - x_1) \), to find the equations of the lines. Here, \( (x_1, y_1) = (4, 6) \) is the point through which the lines pass.

Step 5 ：The equation of the line that is perpendicular to the given line and passes through the point \( (4,6) \) is \( \boxed{4.5x + y = 24} \).

Step 6 ：The equation of the line that is parallel to the given line and passes through the point \( (4,6) \) is \( \boxed{0.222x - y = -5.111} \).