The paths \(x - 3y = 2\) and \(-2x + 6y = 5\) represent straight lines. Do the paths cross each other?
No. They are parallel and distinct.
Step 1: Write the first line equation:
\(x - 3y = 2\)
Step 2: Multiply the whole equation by \(-2\) so that the \(x\)-term looks like the second equation:
\(-2) \times (x - 3y) = (-2) \times 2\)
\(-2x + 6y = -4\)
Step 3: Now compare with the second line:
\(-2x + 6y = 5\)
Step 4: The left-hand sides are the same (\(-2x + 6y\)).
But the right-hand sides are different: one is \(-4\) and the other is \(5\).
Step 5: This means both lines have the same slope, so they are parallel.
Since the constants are not equal, they are not the same line — they are distinct.
Final Answer: Parallel and distinct lines never meet. So the paths do not cross each other.