Exercise 2: Recognizing the No-Solution Case — Possible Solution
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GIVEN
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4x + 6y = 10
2x + 3y = 8

STEP 1: ELIMINATING x
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Using the row operation R2 -> R2 - (1/2) R1 (since row 2's leading
coefficient, 2, is half of row 1's leading coefficient, 4):

  New row 2, column 1: 2 - (1/2)(4) = 0
  New row 2, column 2: 3 - (1/2)(6) = 0
  New row 2, RHS:      8 - (1/2)(10) = 8 - 5 = 3

Resulting row: [0, 0 | 3] -- meaning 0x + 0y = 3.

STEP 2: WHAT THIS ROW MEANS
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0x + 0y = 3 is never true, for any x or y whatsoever. Per this
chapter's own classification, a row that's entirely zero on the left
but nonzero on the right (here, 3) means this system has NO SOLUTION -
this is the inconsistent case, not the infinite-solutions case (which
would instead require the right-hand side to also come out to 0).

STEP 3: THE GEOMETRIC PICTURE
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Dividing the first equation by 2 gives 2x + 3y = 5, directly
comparable to the second equation, 2x + 3y = 8. Both equations have
the exact same left-hand side (the same slope, since they're the same
coefficients up to scaling) but different right-hand sides - these
are two distinct PARALLEL lines. Parallel, non-identical lines never
intersect, which is exactly why there's no (x, y) pair that satisfies
both equations at once.

WHY THIS WORKS AS AN ANSWER
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The elimination is carried out using this chapter's own row-operation
procedure, the resulting all-zero-left/nonzero-right row is matched
to the specific "no solution" case (not the superficially similar
"infinite solutions" case) from this chapter's own classification, and
the geometric explanation is grounded in comparing the two equations'
actual coefficients rather than simply asserting the lines are
parallel.