To solve for the equation of the line [tex]\( WZ \)[/tex] that passes through the point [tex]\((-6, -1)\)[/tex] and is parallel to the line [tex]\( XY \)[/tex] with the equation [tex]\( y = \frac{2}{3}x - 5 \)[/tex], we need to follow these steps:
1. Identify the Slope of the Given Line:
The given line [tex]\( XY \)[/tex] has the equation [tex]\( y = \frac{2}{3}x - 5 \)[/tex]. From this equation, we can see that the slope ([tex]\( m \)[/tex]) of the line is [tex]\( \frac{2}{3} \)[/tex].
2. Use the Point-Slope Form of the Equation:
Since lines [tex]\( XY \)[/tex] and [tex]\( WZ \)[/tex] are parallel, they have the same slope. 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.
3. Substitute the Point and Slope into the Point-Slope Form:
The point [tex]\( WZ \)[/tex] passes through is [tex]\((-6, -1)\)[/tex], and the slope is [tex]\( \frac{2}{3} \)[/tex]. Therefore, we substitute these values into the point-slope form:
[tex]\[
y - (-1) = \frac{2}{3}(x - (-6))
\][/tex]
4. Simplify the Equation:
Simplifying the equation above, we get:
[tex]\[
y + 1 = \frac{2}{3}(x + 6)
\][/tex]
Thus, the correct equation for [tex]\( WZ \)[/tex] in point-slope form, passing through the point [tex]\((-6, -1)\)[/tex] with the same slope as [tex]\( XY \)[/tex] ([tex]\(\frac{2}{3}\)[/tex]), is:
[tex]\[
y - (-1) = \frac{2}{3}(x - (-6))
\][/tex]
Comparing this with the given options, the correct choice is:
[tex]\[
\boxed{y-(-1)=\frac{2}{3}(x-(-6))}
\][/tex]
