Consider the system of equations:2/3x + 4/3y = 5/3 + k/3x3/2y - 1/2 = 5/4x - 3/2yIf k is a...

1. SIMPLIFY the equations to standard form

• First equation: \(\frac{2}{3}\mathrm{x} + \frac{4}{3}\mathrm{y} = \frac{5}{3} + \frac{\mathrm{k}}{3}\mathrm{x}\)

Move the k term to the left side:

\(\frac{2}{3}\mathrm{x} - \frac{\mathrm{k}}{3}\mathrm{x} + \frac{4}{3}\mathrm{y} = \frac{5}{3}\)

Factor out the x terms:

\(\frac{2-\mathrm{k}}{3}\mathrm{x} + \frac{4}{3}\mathrm{y} = \frac{5}{3}\)

Multiply everything by 3 to clear fractions:

\((2-\mathrm{k})\mathrm{x} + 4\mathrm{y} = 5\)

• Second equation: \(\frac{3}{2}\mathrm{y} - \frac{1}{2} = \frac{5}{4}\mathrm{x} - \frac{3}{2}\mathrm{y}\)

Collect y terms on the left:

\(\frac{3}{2}\mathrm{y} + \frac{3}{2}\mathrm{y} = \frac{5}{4}\mathrm{x} + \frac{1}{2}\)

This gives us: \(3\mathrm{y} = \frac{5}{4}\mathrm{x} + \frac{1}{2}\)

Multiply by 4 to clear fractions:

\(12\mathrm{y} = 5\mathrm{x} + 2\)

Rearrange to standard form: \(5\mathrm{x} - 12\mathrm{y} = -2\)

2. INFER the condition for no solution

• Our system is now:
  • \((2-\mathrm{k})\mathrm{x} + 4\mathrm{y} = 5\)
  • \(5\mathrm{x} - 12\mathrm{y} = -2\)
• For no solution, we need parallel lines (same slope, different y-intercepts)
• This means: coefficient ratios must be equal, but constant ratios must be different

3. SIMPLIFY to find k

Set up the coefficient ratio equation:

\(\frac{2-\mathrm{k}}{5} = \frac{4}{-12}\)

SIMPLIFY the right side:

\(\frac{4}{-12} = -\frac{1}{3}\)

So: \(\frac{2-\mathrm{k}}{5} = -\frac{1}{3}\)

Cross multiply:

\(3(2-\mathrm{k}) = 5(-1)\)

\(6 - 3\mathrm{k} = -5\)

\(-3\mathrm{k} = -11\)

\(\mathrm{k} = \frac{11}{3} = 3\frac{2}{3}\) (use calculator)

4. APPLY CONSTRAINTS to verify no solution

• Check that constant ratios are different:
  • Coefficient ratio: \(-\frac{1}{3}\)
  • Constant ratio: \(\frac{5}{-2} = -\frac{5}{2}\)
  • Since \(-\frac{1}{3} \neq -\frac{5}{2}\), we confirmed no solution exists ✓

Answer: C) \(3\frac{2}{3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students make arithmetic errors when manipulating fractions or cross multiplying.

For example, when solving \(\frac{2-\mathrm{k}}{5} = -\frac{1}{3}\), they might incorrectly cross multiply as \(3(2-\mathrm{k}) = 5(1)\) instead of \(3(2-\mathrm{k}) = 5(-1)\), leading to \(6-3\mathrm{k} = 5\), so \(\mathrm{k} = \frac{1}{3}\). Since this doesn't match any answer choice exactly, this leads to confusion and guessing.

Second Most Common Error:

Conceptual confusion about no solution conditions: Students remember that "no solution" involves parallel lines but mix up whether coefficients should be equal or different.

They might set coefficients to be different: \(\frac{2-\mathrm{k}}{5} \neq \frac{4}{-12}\), leading them away from the systematic approach entirely. This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem requires careful fraction manipulation combined with understanding of when linear systems have no solution. The extensive algebraic simplification creates multiple opportunities for computational errors, while the conceptual requirement demands precise recall of parallel line conditions.