Answer:
Step-by-step explanation:
To prove that the lines represented by the equations ax + by + c = 0 and px + qy + r = 0 are parallel if and only if aq = bp, we can use the concept of parallel lines and their slopes.
1. **Determine the Slopes of the Lines:**
- For the equation ax + by + c = 0, the slope is given by -a/b.
- Similarly, for px + qy + r = 0, the slope is -p/q.
2. **Lines are Parallel if Slopes are Equal:**
- Two lines are parallel if their slopes are equal.
- Therefore, the lines represented by the given equations are parallel if -a/b = -p/q.
3. **Prove aq = bp:**
- From the equation of the first line, we have b = -c/a.
- Substituting b = -c/a into the condition -a/b = -p/q gives aq = bp, which completes the proof.
By establishing that the slopes of the lines are equal, we show that the lines are parallel. The condition aq = bp confirms this relationship, demonstrating that the lines are parallel if and only if aq = bp.
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