4x - 9y = 9y + 5hy = 2 + 4xIn the given system of equations, h is a constant....

1. TRANSLATE the equations into standard form

• Given equations:
  • \(\mathrm{4x - 9y = 9y + 5}\)
  • \(\mathrm{hy = 2 + 4x}\)

• TRANSLATE to standard form \(\mathrm{Ax + By = C}\):

First equation: \(\mathrm{4x - 9y = 9y + 5}\)
Subtract 9y from both sides: \(\mathrm{4x - 18y = 5}\)

Second equation: \(\mathrm{hy = 2 + 4x}\)
Rearrange: \(\mathrm{-4x + hy = 2}\)

2. INFER what "no solution" means

• No solution occurs when lines are parallel but not identical
• This means coefficients are proportional: \(\mathrm{A_1/A_2 = B_1/B_2}\)
• But constants are not proportional by the same ratio: \(\mathrm{A_1/A_2 ≠ C_1/C_2}\)

3. SIMPLIFY to find the value of h

• From our standard forms:
  • \(\mathrm{4x - 18y = 5}\) (coefficients: 4, -18, constant: 5)
  • \(\mathrm{-4x + hy = 2}\) (coefficients: -4, h, constant: 2)

• Set up proportion for parallel lines:
  \(\mathrm{4/(-4) = (-18)/h}\)

• SIMPLIFY:
  \(\mathrm{-1 = -18/h}\)
  \(\mathrm{h = 18}\)

4. Verify the answer

• With \(\mathrm{h = 18}\): coefficient ratios are \(\mathrm{4/(-4) = -1}\) and \(\mathrm{-18/18 = -1}\) ✓
• Constant ratio: \(\mathrm{5/2 = 2.5 ≠ -1}\) ✓
• Lines are parallel but distinct = no solution ✓

Answer: D. 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to rewrite equations in standard form correctly, especially with the first equation that has y-terms on both sides.

They might incorrectly write \(\mathrm{4x - 9y = 9y + 5}\) as \(\mathrm{4x - 9y = 5}\), missing the need to combine like terms. This leads to wrong coefficients and an incorrect value for h.

This may lead them to select Choice A (-9) or get stuck and guess.

Second Most Common Error:

Missing conceptual knowledge: Students don't understand what "no solution" means for a system of equations.

They might try to solve the system directly without recognizing that they need to find when the lines are parallel. This leads to confusion about what h should equal.

This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem requires understanding both the algebraic manipulation to get standard form AND the geometric concept that parallel lines create systems with no solution. Students who miss either piece will struggle significantly.