For what values of a and b does the pair 3x + 4y = 12 and (a + b)x + 2(a − b)y = 24 have infinitely many solutions?

1. 1 · Write what is given and what is to be found. a₁ = 3, b₁ = 4, c₁ = 12 and a₂ = a + b, b₂ = 2(a − b), c₂ = 24. To find: a and b.
2. 2 · Write the condition. For infinitely many solutions: a₁/a₂ = b₁/b₂ = c₁/c₂.
3. 3 · Write the ratios. 3/(a + b) = 4/[2(a − b)] = 12/24 = 1/2
4. 4 · Form two equations.
   • 3/(a + b) = 1/2 gives a + b = 6
   • 4/[2(a − b)] = 1/2 gives 2(a − b) = 8, so a − b = 4
5. 5 · Solve. Adding: 2a = 10, so a = 5. Then b = 6 − 5 = 1.
6. 6 · Check and box the answer. The second equation becomes 6x + 8y = 24, which is 2 × (3x + 4y = 12). Therefore a = 5 and b = 1.

Scoring tip: Use the ratio with numbers only (here 12/24 = 1/2) as the target value. It turns one hard equation into two easy ones.