19) through: (-5,-5) perp.to y=-(5)/(2)x A) y=-(1)/(5)x-3 B) y=-3x+(2)/(5) C) y=(1)/(5)x-3 D) y=(2)/(5)x-3

To find the equation of the line that passes through the point $(-5,-5)$ and is perpendicular to the line $y=-\frac{5}{2}x$, we need to determine the slope of the perpendicular line.<br /><br />The slope of the given line $y=-\frac{5}{2}x$ is $-\frac{5}{2}$. The slope of a line perpendicular to this line is the negative reciprocal of $-\frac{5}{2}$, which is $\frac{2}{5}$.<br /><br />Now, we can use the point-slope form of a linear equation to find the equation of the perpendicular line. The point-slope form is given by:<br /><br />$y - y_1 = m(x - x_1)$<br /><br />where $(x_1, y_1)$ is a point on the line and $m$ is the slope.<br /><br />Substituting the point $(-5,-5)$ and the slope $\frac{2}{5}$ into the point-slope form, we get:<br /><br />$y - (-5) = \frac{2}{5}(x - (-5))$<br /><br />Simplifying this equation, we have:<br /><br />$y + 5 = \frac{2}{5}(x + 5)$<br /><br />Multiplying both sides by 5 to eliminate the fraction, we get:<br /><br />$5y + 25 = 2(x + 5)$<br /><br />Expanding and simplifying further, we have:<br /><br />$5y + 25 = 2x + 10$<br /><br />$5y = 2x - 15$<br /><br />Dividing both sides by 5, we get:<br /><br />$y = \frac{2}{5}x - 3$<br /><br />Therefore, the equation of the line that passes through the point $(-5,-5)$ and is perpendicular to the line $y=-\frac{5}{2}x$ is $y = \frac{2}{5}x - 3$.<br /><br />The correct answer is D) $y = \frac{2}{5}x - 3$.