4) through: $$ (5,-4) $$, perp. to $$ y=\frac{5}{2} x+3 $$

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To find the equation of a line that passes through the point $$ (5, -4) $$ and is perpendicular to the line given by the equation $$ y = \frac{5}{2}x + 3 $$, we first need to determine the slope of the given line.

The slope of the line $$ y = \frac{5}{2}x + 3 $$ is $$ \frac{5}{2} $$.

For a line to be perpendicular to another, the product of their slopes must equal $$-1$$. Therefore, if the slope of the given line is $$ m_1 = \frac{5}{2} $$, the slope $$ m_2 $$ of the line we want to find can be calculated as follows:

$$
m_1 \cdot m_2 = -1
$$

Substituting $$ m_1 $$:

$$
\frac{5}{2} \cdot m_2 = -1
$$

To find $$ m_2 $$, we solve for it:

$$
m_2 = -\frac{2}{5}
$$

Now that we have the slope of the line we want to find, we can use the point-slope form of the equation of a line, which is given by:

$$
y - y_1 = m(x - x_1)
$$

where $$ (x_1, y_1) $$ is the point the line passes through, and $$ m $$ is the slope. Substituting $$ (x_1, y_1) = (5, -4) $$ and $$ m = -\frac{2}{5} $$:

$$
y - (-4) = -\frac{2}{5}(x - 5)
$$

This simplifies to:

$$
y + 4 = -\frac{2}{5}(x - 5)
$$

Now, distribute the slope on the right side:

$$
y + 4 = -\frac{2}{5}x + 2
$$

Next, isolate $$ y $$ by subtracting 4 from both sides:

$$
y = -\frac{2}{5}x + 2 - 4
$$

$$
y = -\frac{2}{5}x - 2
$$

Thus, the equation of the line that passes through the point $$ (5, -4) $$ and is perpendicular to the line $$ y = \frac{5}{2}x + 3 $$ is:

$$
\boxed{y = -\frac{2}{5}x - 2}
$$
The equation of the line is $$ y = -\frac{2}{5}x - 2 $$.

4) through: \( (5,-4) \), perp. to \( y=\frac{5}{2} x+3 \)

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