To find the equation of the line [tex]\( YZ \)[/tex] that is parallel to [tex]\( WX \)[/tex] and passes through the point [tex]\((-1, 5)\)[/tex], we'll use the following steps:
1. Identify the slope of line [tex]\( WX \)[/tex]:
The slope-intercept form of a line equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. The equation for [tex]\( WX \)[/tex] is given as:
[tex]\[
y = \frac{1}{4} x - 6
\][/tex]
From this equation, the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{4} \)[/tex].
2. Determine the slope of line [tex]\( YZ \)[/tex]:
Since lines that are parallel have the same slope, the slope of line [tex]\( YZ \)[/tex] is also [tex]\( \frac{1}{4} \)[/tex].
3. Use the point-slope form to find the equation of line [tex]\( YZ \)[/tex]:
The point-slope form of a line equation is:
[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. For line [tex]\( YZ \)[/tex], we use the point [tex]\((-1, 5)\)[/tex] and the slope [tex]\( \frac{1}{4} \)[/tex].
Substituting these values into the point-slope form, we get:
[tex]\[
y - 5 = \frac{1}{4}(x - (-1))
\][/tex]
Simplifying inside the parentheses, we get:
[tex]\[
y - 5 = \frac{1}{4}(x + 1)
\][/tex]
This is the equation for the line [tex]\( YZ \)[/tex]. Therefore, the correct option is:
[tex]\[
y - 5 = \frac{1}{4}(x - (-1))
\][/tex]
