4x - 6y = 10y + 2ty = 1/2 + 2xIn the given system of equations, t is a constant....

1. TRANSLATE the problem information

• Given system:
  • \(\mathrm{4x - 6y = 10y + 2}\)
  • \(\mathrm{ty = \frac{1}{2} + 2x}\)
  • t is a constant
  • System has no solution

• What we need: Find the value of t

2. INFER the approach

• For a system to have no solution, we need an inconsistent equation (like \(\mathrm{0 = 3}\))
• This happens when we eliminate all variables but get a false statement
• Strategy: Manipulate equations to eliminate variables and find when we get a contradiction

3. SIMPLIFY both equations to standard form

• First equation: \(\mathrm{4x - 6y = 10y + 2}\)
  • Move all y terms to left: \(\mathrm{4x - 6y - 10y = 2}\)
  • Combine: \(\mathrm{4x - 16y = 2}\)

• Second equation: \(\mathrm{ty = \frac{1}{2} + 2x}\)
  • Rearrange: \(\mathrm{ty - 2x = \frac{1}{2}}\)
  • Standard form: \(\mathrm{-2x + ty = \frac{1}{2}}\)
  • Multiply by 2 to eliminate fraction: \(\mathrm{-4x + 2ty = 1}\)

4. SIMPLIFY by eliminating x

• Our system now:
  • \(\mathrm{4x - 16y = 2}\)
  • \(\mathrm{-4x + 2ty = 1}\)

• Add the equations:
  \(\mathrm{(4x - 16y) + (-4x + 2ty) = 2 + 1}\)
  \(\mathrm{-16y + 2ty = 3}\)
  \(\mathrm{y(-16 + 2t) = 3}\)

5. INFER the condition for no solution

• For no solution, we need: \(\mathrm{coefficient\ of\ y = 0}\) AND \(\mathrm{constant \neq 0}\)
• Set coefficient equal to zero: \(\mathrm{-16 + 2t = 0}\)
• Solve: \(\mathrm{2t = 16}\), so \(\mathrm{t = 8}\)
• Check: When \(\mathrm{t = 8}\), we get \(\mathrm{y(0) = 3}\), or \(\mathrm{0 = 3}\) ✓ (contradiction)

Answer: 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding what "no solution" means for a system of equations

Students often think "no solution" just means the equations don't intersect, but they don't know how to translate this into algebraic conditions. They might try to solve for x and y directly, get confused when elimination doesn't work cleanly, and then guess among answer choices.

This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Inadequate SIMPLIFY execution: Making algebraic mistakes during equation manipulation

Students might correctly understand the approach but make errors when combining like terms (especially with the \(\mathrm{4x - 6y = 10y + 2}\) equation) or when handling the fraction in the second equation. A common mistake is writing \(\mathrm{4x - 6y - 10y}\) as \(\mathrm{4x - 4y}\) instead of \(\mathrm{4x - 16y}\).

This leads to getting the wrong coefficient equation and selecting an incorrect answer.

The Bottom Line:

This problem requires understanding both the algebraic technique of elimination AND the conceptual meaning of "no solution" as a contradiction. Students who focus only on mechanical manipulation without understanding the underlying logic will struggle to connect their algebra to the question being asked.