Question
Amy is helping plan her school's new basketball court. The west edge of the basketball court is located on the line y = 5x + 2. The east edge cannot intersect with the west edge. On which line could the east edge be located?
−y − 5x = 100
y + 5x = 100
−5x − y = 50
5x − y = 50
Answers
GPT-4o mini
To determine which line could represent the east edge of the basketball court, we need to find a line that does not intersect with the west edge, represented by the equation \(y = 5x + 2\).
First, let’s rewrite the given options in the slope-intercept form (y = mx + b):
1. **Option 1:** \(-y - 5x = 100\)
- Rearranging gives us:
\(-y = 5x + 100 \)
\(y = -5x - 100\)
The slope is \(-5\).
2. **Option 2:** \(y + 5x = 100\)
- Rearranging gives us:
\(y = -5x + 100\)
The slope is \(-5\).
3. **Option 3:** \(-5x - y = 50\)
- Rearranging gives us:
\(-y = 5x + 50\)
\(y = -5x - 50\)
The slope is \(-5\).
4. **Option 4:** \(5x - y = 50\)
- Rearranging gives us:
\(-y = -5x + 50\)
\(y = 5x - 50\)
The slope is \(5\).
Next, let’s analyze the slopes. The west edge of the basketball court has a slope of \(5\). For the east edge to avoid intersecting with the west edge, it should have a different slope or be positioned appropriately.
The options with a slope of \(-5\) are:
- Option 1: \(y = -5x - 100\)
- Option 2: \(y = -5x + 100\)
- Option 3: \(y = -5x - 50\)
These lines are all negatively sloped and will never intersect with the positively sloped line \(y = 5x + 2\).
The last option:
- Option 4: \(y = 5x - 50\) has the same slope of \(5\) as the west edge, so it will intersect with the west edge.
Thus, the east edge must be on a line that does not share the same slope or intersect. Therefore, the east edge of the basketball court could be located on any of the following lines:
**y = -5x - 100, y = -5x + 100, y = -5x - 50.**
Since we are looking for a specific answer from the given choices, both \(-y - 5x = 100\), \(y + 5x = 100\), and \(-5x - y = 50\) would be valid, but one specific option presented should be selected.
The answer is:
**5x - y = 50.**
First, let’s rewrite the given options in the slope-intercept form (y = mx + b):
1. **Option 1:** \(-y - 5x = 100\)
- Rearranging gives us:
\(-y = 5x + 100 \)
\(y = -5x - 100\)
The slope is \(-5\).
2. **Option 2:** \(y + 5x = 100\)
- Rearranging gives us:
\(y = -5x + 100\)
The slope is \(-5\).
3. **Option 3:** \(-5x - y = 50\)
- Rearranging gives us:
\(-y = 5x + 50\)
\(y = -5x - 50\)
The slope is \(-5\).
4. **Option 4:** \(5x - y = 50\)
- Rearranging gives us:
\(-y = -5x + 50\)
\(y = 5x - 50\)
The slope is \(5\).
Next, let’s analyze the slopes. The west edge of the basketball court has a slope of \(5\). For the east edge to avoid intersecting with the west edge, it should have a different slope or be positioned appropriately.
The options with a slope of \(-5\) are:
- Option 1: \(y = -5x - 100\)
- Option 2: \(y = -5x + 100\)
- Option 3: \(y = -5x - 50\)
These lines are all negatively sloped and will never intersect with the positively sloped line \(y = 5x + 2\).
The last option:
- Option 4: \(y = 5x - 50\) has the same slope of \(5\) as the west edge, so it will intersect with the west edge.
Thus, the east edge must be on a line that does not share the same slope or intersect. Therefore, the east edge of the basketball court could be located on any of the following lines:
**y = -5x - 100, y = -5x + 100, y = -5x - 50.**
Since we are looking for a specific answer from the given choices, both \(-y - 5x = 100\), \(y + 5x = 100\), and \(-5x - y = 50\) would be valid, but one specific option presented should be selected.
The answer is:
**5x - y = 50.**