To determine the slope of the line passing through the points [tex]\( U(1, -4) \)[/tex] and [tex]\( K(-2, 8) \)[/tex], we use the slope formula. The formula for the slope [tex]\( m \)[/tex] between any two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, [tex]\( (x_1, y_1) = (1, -4) \)[/tex] and [tex]\( (x_2, y_2) = (-2, 8) \)[/tex].
Plugging these coordinates into the slope formula gives:
[tex]\[
m = \frac{8 - (-4)}{-2 - 1}
\][/tex]
First, simplify the numerator and the denominator:
[tex]\[
m = \frac{8 + 4}{-2 - 1}
\][/tex]
This simplifies to:
[tex]\[
m = \frac{12}{-3}
\][/tex]
Finally, divide 12 by -3:
[tex]\[
m = -4
\][/tex]
Therefore, the slope of the line passing through the points [tex]\( U(1, -4) \)[/tex] and [tex]\( K(-2, 8) \)[/tex] is:
[tex]\[
\boxed{-4}
\][/tex]
