AlgebraHardConstant for a dependent system
Algebra practice question
2x + 3y = 12
4x + 6y = k
For what value of k does the system have infinitely many solutions?
Answer: 24
A system has infinitely many solutions when the two equations describe the same line. The second equation's left side, 4x + 6y, is exactly 2 times the first's, 2x + 3y. So the right side must be 2 times 12 as well: k = 24.
Common wrong answers
- If you answered 6
- 6 is 12 halved. The second equation's left side is DOUBLE the first's, so the right side doubles too: 24.
- If you answered 12
- That keeps the right side unchanged while the left side doubled, giving 4x + 6y = 12 — a parallel line, so NO solutions. The whole equation must be scaled by 2.
- If you answered 48
- 48 is 12 × 4, but the scale factor between 2x + 3y and 4x + 6y is 2, not 4.
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