For the following equations, what are the values of a and b to have infinitely many solutions? ax + by = 2 3x - (5 - 2a)y = 6

Correct — A. A pair of linear equations has infinitely many solutions when the two describe the same line, that is when a₁⁄a₂ = b₁⁄b₂ = c₁⁄c₂.

ax + by = 2 gives (a, b, 2)
3x − (5 − 2a)y = 6 gives (3, −(5 − 2a), 6)

The constants are fully known, so start there:
c₁⁄c₂ = 2⁄6 = 1⁄3

x-terms: a⁄3 = 1⁄3, so a = 1
y-terms: with a = 1, −(5 − 2a) = −3, so b⁄(−3) = 1⁄3 and b = −1

Check: x − y = 2 and 3x − 3y = 6 are the same line → option (a).

Why the others are wrong

• (b)
  a = - 1,b = - 1 — a = −1 makes the x-ratio −1⁄3 against the constants' 1⁄3, so the equations are not multiples of each other. −x − y = 2 and 3x − 7y = 6 meet at a single point.
• (c)
  a = - 1,b = 1 — The x-term fails again: a = −1 gives −1⁄3, not 1⁄3. Nor is b right: with a = −1 the second equation is 3x − 7y = 6, so b = 1 gives a y-ratio of −1⁄7.
• (d)
  a = 1,b = 1 — a = 1 is right, but b = 1 makes the y-ratio −1⁄3 instead of 1⁄3. The lines x + y = 2 and x − y = 2 then cross at one point rather than coinciding.

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

If a₁⁄a₂ ≠ b₁⁄b₂ the lines cross once — a unique solution. If a₁⁄a₂ = b₁⁄b₂ but c₁⁄c₂ differs, the lines are parallel and there is no solution.

If all three are equal the equations are multiples of one another, so they draw the same line and every point on it solves both. Start from whichever ratio is already fully known: here 2 and 6 fix it at 1⁄3 straight away, and each coefficient then falls out.

Key facts

• Infinitely many solutions require a₁⁄a₂ = b₁⁄b₂ = c₁⁄c₂.
• No solution requires a₁⁄a₂ = b₁⁄b₂ ≠ c₁⁄c₂.
• A unique solution requires only a₁⁄a₂ ≠ b₁⁄b₂.
• Here 2⁄6 = 1⁄3 forces a = 1 and b = −1, and both equations reduce to x − y = 2.

Common traps

• Dropping the minus sign in −(5 − 2a) and solving for b as +1.
• Checking only two of the three ratios and accepting a parallel pair by mistake.
• Using the y-ratio before a has been fixed from the x-ratio and the constants.

No directly related past PYQ was found.