To determine the value of [tex]\( d \)[/tex], we can use the formula for the slope of a line that connects two points. The slope [tex]\( m \)[/tex] of a line through points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
In this case, the points are [tex]\((10, 8)\)[/tex] and [tex]\((d, -4)\)[/tex], and we know the slope of the line is 2. So, we can write:
[tex]\[
2 = \frac{-4 - 8}{d - 10}
\][/tex]
Simplifying the numerator on the right-hand side:
[tex]\[
2 = \frac{-12}{d - 10}
\][/tex]
Next, we will solve for [tex]\( d \)[/tex] by isolating the variable. Start by multiplying both sides of the equation by [tex]\( d - 10 \)[/tex]:
[tex]\[
2(d - 10) = -12
\][/tex]
Distribute the 2 on the left-hand side:
[tex]\[
2d - 20 = -12
\][/tex]
Add 20 to both sides to isolate the term with [tex]\( d \)[/tex]:
[tex]\[
2d - 20 + 20 = -12 + 20
\][/tex]
[tex]\[
2d = 8
\][/tex]
Finally, divide both sides by 2 to solve for [tex]\( d \)[/tex]:
[tex]\[
d = \frac{8}{2}
\][/tex]
[tex]\[
d = 4
\][/tex]
Thus, the value of [tex]\( d \)[/tex] is [tex]\(\boxed{4}\)[/tex].
