To determine which equation could represent the east edge of the basketball court, let's analyze the given options. We need to ensure that the east edge does not intersect with the west edge, whose equation is [tex]\( y = 5x + 2 \)[/tex].
We will convert each given equation into the slope-intercept form [tex]\( y = mx + b \)[/tex] and compare their slopes to the slope of the west edge. If a line has the same slope but different y-intercepts, it is parallel to the west edge and will not intersect it.
1. Option 1: [tex]\( -y - 5x = 100 \)[/tex]
- Rearrange to isolate [tex]\( y \)[/tex]:
[tex]\[
-y - 5x = 100 \\
y + 5x = -100 \\
y = -5x - 100
\][/tex]
- The slope is [tex]\( m = -5 \)[/tex].
2. Option 2: [tex]\( y + 5x = 100 \)[/tex]
- It is already in slope-intercept form:
[tex]\[
y = -5x + 100
\][/tex]
- The slope is [tex]\( m = -5 \)[/tex].
3. Option 3: [tex]\( -5x - y = 50 \)[/tex]
- Rearrange to isolate [tex]\( y \)[/tex]:
[tex]\[
-5x - y = 50 \\
5x + y = -50 \\
y = -5x - 50
\][/tex]
- The slope is [tex]\( m = -5 \)[/tex].
4. Option 4: [tex]\( 5x - y = 50 \)[/tex]
- Rearrange to isolate [tex]\( y \)[/tex]:
[tex]\[
5x - y = 50 \\
-y = 50 - 5x \\
y = 5x - 50
\][/tex]
- The slope is [tex]\( m = 5 \)[/tex], which is the same as the slope of the west edge.
Since parallel lines do not intersect, only lines with different slopes might intersect. The west edge of the basketball court has a slope of [tex]\( 5 \)[/tex] in the equation [tex]\( y = 5x + 2 \)[/tex]. Therefore, the equation of the east edge that does not intersect with the west edge and has a different slope is:
[tex]\[ \boxed{5x - y = 50} \][/tex]
