Question

parallel lines have different slopes
the point of intersection is called the solution
no solution means the slopes were the same
infinite solutions means the lines are the same
one solution means equal slopes
the no solution example, both slopes are negative

Explanation:

To solve this, we analyze each statement about linear equations and their slopes:

1. Parallel lines have different slopes

• Fact: Parallel lines have equal slopes (and different y - intercepts). So this statement is False.

2. The point of intersection is called the solution

• Fact: For a system of linear equations, the point where the lines intersect (if they do) is the solution (satisfies both equations). So this statement is True.

3. No solution means the slopes were the same

• Fact: A system with no solution has parallel lines, which have equal slopes (and different y - intercepts). So this statement is True.

4. Infinite solutions means the lines are the same

• Fact: If two lines are identical (same slope and same y - intercept), they overlap completely, giving infinite solutions. So this statement is True.

5. One solution means equal slopes

• Fact: If slopes are equal, lines are parallel (no solution) or coincident (infinite solutions). For one solution, slopes must be different (lines intersect at one point). So this statement is False.

6. In the no - solution example, both slopes are negative

• Fact: Parallel lines (no solution) can have any equal slopes (both positive, both negative, or zero/undefined in special cases). So this is a possible case (e.g., \(y=-2x + 1\) and \(y=-2x + 3\) are parallel with negative slopes and no solution), so this statement is True (as a valid example of no - solution with negative slopes).

To match (assuming the right - hand side is for marking true/false or correcting, but based on analysis):

• "Parallel lines have different slopes" → False (correct: equal slopes)
• "The point of intersection is called the solution" → True
• "No solution means the slopes were the same" → True
• "Infinite solutions means the lines are the same" → True
• "One solution means equal slopes" → False (correct: different slopes)
• "In the no - solution example, both slopes are negative" → True (valid example)

If the task is to identify correct statements: The correct ones are "The point of intersection is called the solution", "No solution means the slopes were the same", "Infinite solutions means the lines are the same", "In the no - solution example, both slopes are negative". The incorrect ones are "Parallel lines have different slopes" and "One solution means equal slopes".

Answer:

To solve this, we analyze each statement about linear equations and their slopes:

1. Parallel lines have different slopes

• Fact: Parallel lines have equal slopes (and different y - intercepts). So this statement is False.

2. The point of intersection is called the solution

• Fact: For a system of linear equations, the point where the lines intersect (if they do) is the solution (satisfies both equations). So this statement is True.

3. No solution means the slopes were the same

• Fact: A system with no solution has parallel lines, which have equal slopes (and different y - intercepts). So this statement is True.

4. Infinite solutions means the lines are the same

• Fact: If two lines are identical (same slope and same y - intercept), they overlap completely, giving infinite solutions. So this statement is True.

5. One solution means equal slopes

• Fact: If slopes are equal, lines are parallel (no solution) or coincident (infinite solutions). For one solution, slopes must be different (lines intersect at one point). So this statement is False.

6. In the no - solution example, both slopes are negative

• Fact: Parallel lines (no solution) can have any equal slopes (both positive, both negative, or zero/undefined in special cases). So this is a possible case (e.g., \(y=-2x + 1\) and \(y=-2x + 3\) are parallel with negative slopes and no solution), so this statement is True (as a valid example of no - solution with negative slopes).

To match (assuming the right - hand side is for marking true/false or correcting, but based on analysis):

• "Parallel lines have different slopes" → False (correct: equal slopes)
• "The point of intersection is called the solution" → True
• "No solution means the slopes were the same" → True
• "Infinite solutions means the lines are the same" → True
• "One solution means equal slopes" → False (correct: different slopes)
• "In the no - solution example, both slopes are negative" → True (valid example)

If the task is to identify correct statements: The correct ones are "The point of intersection is called the solution", "No solution means the slopes were the same", "Infinite solutions means the lines are the same", "In the no - solution example, both slopes are negative". The incorrect ones are "Parallel lines have different slopes" and "One solution means equal slopes".