Which of the following is the sum of the values of a and b if the equations 2x + y = a, 8x + by = 12 have infinite solutions?

Correct — C. Infinite solutions means the two equations describe the same line, so all three coefficient ratios must match.

Condition: a₁⁄a₂ = b₁⁄b₂ = c₁⁄c₂ → 2⁄8 = 1⁄b = a⁄12.

2⁄8 = 1⁄4, so the scale factor is 4.
1⁄b = 1⁄4 → b = 4.
a⁄12 = 1⁄4 → a = 3.

Check: multiplying 2x + y = 3 by 4 gives 8x + 4y = 12, which is the second equation exactly.

a + b = 3 + 4 = 7 → option (c).

Why the others are wrong

• (a)
  16 — 16 is 12 + 4 — correct on b, wrong on a. Multiplying 2x + y = a by 4 makes the constant 4a, so 4a = 12 and a = 3; copying 12 across as a is the slip.
• (b)
  9 — 9 needs a = 5 alongside b = 4, and nothing in the ratio chain yields 5. The constant is pinned by the same factor of 4 as the coefficients.
• (d)
  18 — 18 overshoots what the scale factor allows. Once b = 4 is forced by 2⁄8, the constant must satisfy 4a = 12, so a + b cannot exceed 7.

For a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the three ratios decide everything.

a₁⁄a₂ ≠ b₁⁄b₂ gives exactly one solution — the lines cross.
a₁⁄a₂ = b₁⁄b₂ ≠ c₁⁄c₂ gives no solution — the lines are parallel and distinct.
a₁⁄a₂ = b₁⁄b₂ = c₁⁄c₂ gives infinitely many — the lines coincide.

Here the x-coefficients fix the scale factor at 1⁄4, and that single factor then determines both unknowns: one sitting in a y-coefficient and one sitting in a constant.

Key facts

• Infinitely many solutions require a₁⁄a₂ = b₁⁄b₂ = c₁⁄c₂.
• No solution requires a₁⁄a₂ = b₁⁄b₂ ≠ c₁⁄c₂.
• Here b = 4 and a = 3, and 2x + y = 3 multiplied by 4 reproduces 8x + 4y = 12.

Common traps

• Using only the two coefficient ratios, which is the test for parallel lines and gives no solution.
• Inverting the constant ratio to 12⁄a and landing on a = 48.
• Finding a and b correctly and then reporting one of them instead of the sum.

No directly related past PYQ was found.