To find the equation of a line that is parallel to line [tex]\( CD \)[/tex] and passes through the point [tex]\((4, 5)\)[/tex], we follow these steps:
1. Identify the slope of line [tex]\( CD \)[/tex]:
The given equation of the line [tex]\( CD \)[/tex] is [tex]\( y = -2x - 2 \)[/tex]. The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. For the equation [tex]\( y = -2x - 2 \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\(-2\)[/tex].
2. Use the point-slope form of the equation of a line:
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, we use the slope [tex]\( m = -2 \)[/tex] and the point [tex]\((x_1, y_1) = (4, 5)\)[/tex].
3. Substitute the slope and the point into the point-slope form:
[tex]\[
y - 5 = -2(x - 4)
\][/tex]
4. Simplify the equation to the slope-intercept form:
Expand and simplify the equation:
[tex]\[
y - 5 = -2(x - 4)
\][/tex]
Distribute the slope [tex]\(-2\)[/tex] through the parentheses:
[tex]\[
y - 5 = -2x + 8
\][/tex]
Add 5 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -2x + 13
\][/tex]
So, the equation of the line that is parallel to [tex]\( CD \)[/tex] and passes through the point [tex]\((4, 5)\)[/tex] is:
[tex]\[
y = -2x + 13
\][/tex]
Therefore, the correct answer from the given choices is:
[tex]\[
\boxed{y = -2x + 13}
\][/tex]
