Answer:
To find the equation of a line perpendicular to the given line \(y = -\frac{2}{9}x + 4\) that passes through the point (-2, -4), we'll first determine the slope of the perpendicular line.
The given line \(y = -\frac{2}{9}x + 4\) has a slope of -2/9 because it's in the form \(y = mx + b\), where \(m\) is the slope.
For a line to be perpendicular to this line, it must have a slope that is the negative reciprocal of -2/9. The negative reciprocal of a number is obtained by flipping the fraction and changing the sign, so the perpendicular slope is \(m = \frac{9}{2}\).
Now, we have the slope (\(m = \frac{9}{2}\)) and a point (-2, -4). We can use the point-slope form of a line to find the equation:
\(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the given point and \(m\) is the slope.
Plugging in the values:
\(y - (-4) = \frac{9}{2}(x - (-2))\)
\(y + 4 = \frac{9}{2}(x + 2)\)
Now, let's simplify and get the equation in standard form (Ax + By = C):
\(2(y + 4) = 9(x + 2)\)
\(2y + 8 = 9x + 18\)
Subtract 9x and 8 from both sides:
\(2y = 9x + 18 - 8\)
\(2y = 9x + 10\)
Divide both sides by 2 to isolate y:
\(y = \frac{9}{2}x + 5\)
So, the equation of the line perpendicular to \(y = -\frac{2}{9}x + 4\) that passes through (-2, -4) is:
\(y = \frac{9}{2}x + 5\)
So, the correct answer is option (b) \(y = \frac{9}{2}x + 5\).
Explanation: