To determine which line the east edge of the basketball court could be located on so that it does not intersect with the west edge, we first need to understand that the west edge is given by the equation [tex]\( y = -4x \)[/tex]. For the east edge to not intersect with the west edge, the two lines must be parallel. In other words, the slope of the east edge must be the same as the slope of the west edge.
Let's analyze each given line to determine its slope and find out which one is parallel to [tex]\( y = -4x \)[/tex]:
1. Line 1: [tex]\( y - 4x = -200 \)[/tex]
- Rewrite in slope-intercept form: [tex]\( y = 4x - 200 \)[/tex]
- Slope: [tex]\( 4 \)[/tex]
2. Line 2: [tex]\( -4x - y = -50 \)[/tex]
- Rewrite in slope-intercept form: [tex]\( -y = 4x - 50 \)[/tex]
- Multiply both sides by -1 to solve for [tex]\( y \)[/tex]: [tex]\( y = -4x + 50 \)[/tex]
- Slope: [tex]\( -4 \)[/tex]
3. Line 3: [tex]\( 4x - y = -200 \)[/tex]
- Rewrite in slope-intercept form: [tex]\( -y = -4x - 200 \)[/tex]
- Multiply both sides by -1 to solve for [tex]\( y \)[/tex]: [tex]\( y = 4x + 200 \)[/tex]
- Slope: [tex]\( 4 \)[/tex]
4. Line 4: [tex]\( -y + 4x = -50 \)[/tex]
- Rewrite in slope-intercept form: [tex]\( -y = -4x - 50 \)[/tex]
- Multiply both sides by -1 to solve for [tex]\( y \)[/tex]: [tex]\( y = 4x + 50 \)[/tex]
- Slope: [tex]\( 4 \)[/tex]
Out of these equations, only Line 2: [tex]\( -4x - y = -50 \)[/tex] has a slope of [tex]\( -4 \)[/tex], which matches the slope of the west edge [tex]\( y = -4x \)[/tex]. Therefore, this line is parallel to the west edge and does not intersect with it.
Hence, the east edge of the basketball court could be located on the line:
[tex]\[ \boxed{-4x - y = -50} \][/tex]
