To find the equation of line [tex]\( WZ \)[/tex] which passes through the point [tex]\((-6, -1)\)[/tex] and is parallel to line [tex]\( XY \)[/tex] defined by [tex]\( y = \frac{2}{3}x - 5 \)[/tex], we need to follow these steps:
1. Identify the slope of line XY:
The given equation for line [tex]\( XY \)[/tex] is [tex]\( y = \frac{2}{3}x - 5 \)[/tex]. The coefficient of [tex]\( x \)[/tex] in this equation is the slope. Therefore, the slope of line [tex]\( XY \)[/tex] is [tex]\( \frac{2}{3} \)[/tex].
2. Determine the slope of line WZ:
Since [tex]\( WZ \)[/tex] is parallel to [tex]\( XY \)[/tex], it will have the same slope. Hence, the slope of line [tex]\( WZ \)[/tex] is also [tex]\( \frac{2}{3} \)[/tex].
3. Use the point-slope form of the equation of a line:
The point-slope form of a line's equation is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line, and [tex]\( m \)[/tex] is the slope.
4. Substitute the slope and point into the point-slope form:
Here, the slope [tex]\( m \)[/tex] is [tex]\( \frac{2}{3} \)[/tex], and the point [tex]\((-6, -1)\)[/tex] is on the line. Substituting these values, we get:
[tex]\[
y - (-1) = \frac{2}{3}(x - (-6))
\][/tex]
5. Simplify the equation:
Simplifying this equation, we have:
[tex]\[
y + 1 = \frac{2}{3}(x + 6)
\][/tex]
6. Determine the correct option:
Comparing this with the given options:
[tex]\[
y - (-1) = \frac{2}{3}(x - (-6))
\][/tex]
This matches the process where:
[tex]\[
y + 1 = \frac{2}{3}(x + 6)
\][/tex]
Therefore, the correct equation for line [tex]\( WZ \)[/tex] passing through the point [tex]\((-6, -1)\)[/tex] and parallel to line [tex]\( XY \)[/tex] is represented by the option:
[tex]\[ y - (-1) = \frac{2}{3}(x - (-6)) \][/tex]
So, the correct answer is the fourth option:
[tex]\[ \boxed{4} \][/tex]
