Answer :
Sure, let's analyze each statement in detail and determine their validity:
### Statement 15:
"Two lines intersect in at most one point."
In Euclidean geometry, which is the geometry of the flat plane, two distinct lines can intersect in at most one point if they are not parallel. Here are the possible scenarios:
- If two distinct lines are not parallel, they will intersect at exactly one point.
- If two lines are parallel, they will never intersect.
Since these are the only two possibilities in Euclidean geometry, the statement "Two lines intersect in at most one point" is indeed true.
Answer: True
### Statement 16:
"The equation [tex]\(2x - 7 = 5 + x\)[/tex] has exactly one solution."
Let's solve this linear equation step-by-step:
1. Given the equation:
[tex]\[2x - 7 = 5 + x\][/tex]
2. Subtract [tex]\(x\)[/tex] from both sides to isolate [tex]\(x\)[/tex] on one side:
[tex]\[2x - x - 7 = 5\][/tex]
3. Simplify the left-hand side:
[tex]\[x - 7 = 5\][/tex]
4. Add 7 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[x = 12\][/tex]
The equation [tex]\(2x - 7 = 5 + x\)[/tex] has exactly one solution, which is [tex]\(x = 12\)[/tex].
Answer: True
### Summary:
- Statement 15: True
- Statement 16: True
### Statement 15:
"Two lines intersect in at most one point."
In Euclidean geometry, which is the geometry of the flat plane, two distinct lines can intersect in at most one point if they are not parallel. Here are the possible scenarios:
- If two distinct lines are not parallel, they will intersect at exactly one point.
- If two lines are parallel, they will never intersect.
Since these are the only two possibilities in Euclidean geometry, the statement "Two lines intersect in at most one point" is indeed true.
Answer: True
### Statement 16:
"The equation [tex]\(2x - 7 = 5 + x\)[/tex] has exactly one solution."
Let's solve this linear equation step-by-step:
1. Given the equation:
[tex]\[2x - 7 = 5 + x\][/tex]
2. Subtract [tex]\(x\)[/tex] from both sides to isolate [tex]\(x\)[/tex] on one side:
[tex]\[2x - x - 7 = 5\][/tex]
3. Simplify the left-hand side:
[tex]\[x - 7 = 5\][/tex]
4. Add 7 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[x = 12\][/tex]
The equation [tex]\(2x - 7 = 5 + x\)[/tex] has exactly one solution, which is [tex]\(x = 12\)[/tex].
Answer: True
### Summary:
- Statement 15: True
- Statement 16: True
