Given a linear equation in two variables: 5x + 7y - 8 = 0, which of the following linear equations, along with the given equation, forms a system of linear equations having no solution?

Correct — B. A pair of lines has no solution exactly when

a₁⁄a₂ = b₁⁄b₂ ≠ c₁⁄c₂

Set 5x + 7y − 8 = 0 against option (b), 5x + 7y − 16 = 0:
5⁄5 = 1 and 7⁄7 = 1, so the two coefficient ratios agree,
but the constants give −8⁄−16 = 1⁄2, which does not.

Equal slopes with different intercepts means the lines are parallel and distinct, so they never meet → option (b).

Why the others are wrong

• (a)
  10x + 14y - 16 = 0 — This is the given equation doubled: 5⁄10 = 7⁄14 = −8⁄−16 = 1⁄2, all three ratios equal. That is the same line written twice, so it has infinitely many solutions, not none.
• (c)
  5x - 7y - 8 = 0 — The sign of the y term flips: 5⁄5 = 1 but 7⁄(−7) = −1. Unequal coefficient ratios mean the lines cross, giving exactly one solution.
• (d)
  7x + 5y - 8 = 0 — The two coefficients are swapped: 5⁄7 is not 7⁄5, so the slopes differ and the lines intersect at a single point.

A pair of linear equations in two variables is a pair of lines, and there are exactly three outcomes.

Unequal coefficient ratios means the lines cross once, so there is one solution.
All three ratios equal means the two equations are the same line, so there are infinitely many.
Equal coefficient ratios with an unequal constant ratio means parallel but distinct lines, so there is no solution.

Only the third case answers this question, and the test is mechanical: put both equations in ax + by + c = 0 form, then compare a₁⁄a₂, b₁⁄b₂ and c₁⁄c₂ in that order.

Key facts

• a₁⁄a₂ ≠ b₁⁄b₂ gives exactly one solution, because the lines intersect.
• a₁⁄a₂ = b₁⁄b₂ = c₁⁄c₂ gives infinitely many solutions, because the lines coincide.
• a₁⁄a₂ = b₁⁄b₂ ≠ c₁⁄c₂ gives no solution, because the lines are parallel and distinct.
• Both equations must be arranged as ax + by + c = 0 before any ratio is compared.

Common traps

• Picking the multiplied-through equation, which gives infinitely many solutions instead of none.
• Comparing only the x and y coefficients and never checking the constants.
• Comparing ratios before every term has been moved to one side.

No directly related past PYQ was found.