determine the equation of the line passing thru the midpoint between (-5,0) and (0,5) and is perpendicular to the line segment between (-3,0) and (0,5)

To find the equation of the line passing through the midpoint of (-5, 0) and (0, 5), we first need to find the coordinates of the midpoint. The midpoint coordinates can be found using the midpoint formula:

Midpoint = ((\frac{x_1 + x_2}{2}), (\frac{y_1 + y_2}{2}))

Midpoint = ((\frac{-5 + 0}{2}), (\frac{0 + 5}{2}))

Midpoint = (\frac{-5}{2}, \frac{5}{2})

Midpoint = (-\frac{5}{2}, \frac{5}{2})

So, the midpoint of the line passing through (-5, 0) and (0, 5) is (-\frac{5}{2}, \frac{5}{2}) .

Next, we need to determine the slope of the line passing through (-3, 0) and (0, 5):

Slope = \frac{y_2 - y_1}{x_2 - x_1}

Slope = \frac{5 - 0}{0 + 3}

Slope = \frac{5}{3}

The line passing through the midpoint (-\frac{5}{2}, \frac{5}{2}) and perpendicular to the line passing through (-3, 0) and (0, 5) will have a slope that is the negative reciprocal of \frac{5}{3} , which is -\frac{3}{5} .

Now we have the midpoint (-\frac{5}{2}, \frac{5}{2}) and the slope -\frac{3}{5} . We can use the point-slope form of the equation of a line to find the equation:

Point-slope form: (y - y_1) = m(x - x_1)

Substitute the midpoint (-\frac{5}{2}, \frac{5}{2}) and the slope -\frac{3}{5} into the point-slope form:

(y - \frac{5}{2}) = -\frac{3}{5}(x + \frac{5}{2})

Simplify the equation:

5y - \frac{25}{2} = -3x - \frac{15}{2}

5y = -3x + 5

\boxed{5y + 3x = 5}

Therefore, the equation of the line passing through the midpoint between (-5, 0) and (0, 5) and perpendicular to the line passing through (-3, 0) and (0, 5) is 5y + 3x = 5 .