What conclusion can be drawn about the solution of the following system of linear equations in two variables: 3x + 2y = 7 2x + 3y = 7

Correct — C. Compare the coefficient ratios of 3x + 2y = 7 and 2x + 3y = 7 before solving anything.

a₁/a₂ = 3/2 and b₁/b₂ = 2/3
3/2 ≠ 2/3, so the two lines have different slopes

Lines with different slopes meet at exactly one point — a unique solution → option (c)

Solving confirms it:
subtract: x − y = 0
add: 5x + 5y = 14
so x = y = 7/5

Why the others are wrong

• (a)
  More than two solution — More than two solutions is not a case a pair of straight lines can produce. Two lines either cross once, never meet, or coincide and share every point — there is no three-solution or four-solution outcome.
• (b)
  Infinite solutions — Infinite solutions needs a₁/a₂ = b₁/b₂ = c₁/c₂, that is, one equation a plain multiple of the other. This pair fails at the first ratio, since 3/2 ≠ 2/3.
• (d)
  No solution — No solution means parallel lines, which needs a₁/a₂ = b₁/b₂ ≠ c₁/c₂. The slopes here are −3/2 and −2/3, so the lines cross instead of running parallel.

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, three ratios settle the whole question:

a₁/a₂ ≠ b₁/b₂ → the lines intersect once, a unique solution.
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → parallel lines, no solution.
a₁/a₂ = b₁/b₂ = c₁/c₂ → one and the same line, infinitely many solutions.

Test the first pair of ratios before anything else. If they differ you are already done, and the actual values of x and y are never needed to answer.

Key facts

• 3/2 ≠ 2/3, so 3x + 2y = 7 and 2x + 3y = 7 meet at exactly one point.
• The solution is x = y = 7/5, since 3(7/5) + 2(7/5) = 35/5 = 7.
• A pair of linear equations in two variables has one solution, none, or infinitely many — never a finite number greater than one.

Common traps

• Checking a₁/a₂ = b₁/b₂ and forgetting c₁/c₂, the ratio that separates parallel from coincident
• Reading the two equal right-hand sides (7 and 7) as a sign that the equations are equivalent

No directly related past PYQ was found.