2/5x + 7/5y = 2/7 gx + ky = 5/2 In the given system of equations, g and k are...

1. TRANSLATE the problem information

• Given system:
  • \(\frac{2}{5}\mathrm{x} + \frac{7}{5}\mathrm{y} = \frac{2}{7}\)
  • \(\mathrm{gx} + \mathrm{ky} = \frac{5}{2}\)
• The system has infinitely many solutions
• Find: \(\frac{\mathrm{g}}{\mathrm{k}}\)

2. INFER the key relationship

• For a system to have infinitely many solutions, the equations must be equivalent
• This means one equation is a scalar multiple of the other
• So there exists some constant c where: \(\mathrm{gx} + \mathrm{ky} = \mathrm{c}(\frac{2}{5}\mathrm{x} + \frac{7}{5}\mathrm{y})\)

3. SIMPLIFY to set up the proportionality

• If \(\mathrm{gx} + \mathrm{ky} = \mathrm{c}(\frac{2}{5}\mathrm{x} + \frac{7}{5}\mathrm{y})\), then:
  • \(\mathrm{g} = \mathrm{c}(\frac{2}{5}) = \frac{2\mathrm{c}}{5}\)
  • \(\mathrm{k} = \mathrm{c}(\frac{7}{5}) = \frac{7\mathrm{c}}{5}\)
• The right sides must also be equal: \(\frac{5}{2} = \mathrm{c}(\frac{2}{7})\)

4. SIMPLIFY to find the constant c

• From \(\frac{5}{2} = \mathrm{c}(\frac{2}{7})\):
• \(\mathrm{c} = (\frac{5}{2}) \div (\frac{2}{7})\)
  \(\mathrm{c} = (\frac{5}{2}) \times (\frac{7}{2})\)
  \(\mathrm{c} = \frac{35}{4}\)

5. SIMPLIFY to find g and k

• \(\mathrm{g} = \frac{2\mathrm{c}}{5} = \frac{2(\frac{35}{4})}{5} = \frac{70}{20} = \frac{7}{2}\)
• \(\mathrm{k} = \frac{7\mathrm{c}}{5} = \frac{7(\frac{35}{4})}{5} = \frac{245}{20} = \frac{49}{4}\)

6. SIMPLIFY to find the final ratio

• \(\frac{\mathrm{g}}{\mathrm{k}} = (\frac{7}{2}) \div (\frac{49}{4})\)
  \(\frac{\mathrm{g}}{\mathrm{k}} = (\frac{7}{2}) \times (\frac{4}{49})\)
  \(\frac{\mathrm{g}}{\mathrm{k}} = \frac{28}{98} = \frac{2}{7}\)

Answer: 2/7, 0.285, 0.286, .2857

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that "infinitely many solutions" means the equations must be equivalent/proportional.

Instead, they might try to solve the system normally or set up arbitrary relationships between g and k without using the constraint. This leads to confusion and guessing since they can't establish the necessary proportionality relationship.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the proportionality concept but make arithmetic errors when working with complex fractions.

Common mistakes include errors in fraction division (like \((\frac{5}{2}) \div (\frac{2}{7})\)) or mistakes when simplifying the final ratio. This may lead them to get a result like \(\frac{7}{2}\) instead of \(\frac{2}{7}\), essentially getting the reciprocal of the correct answer.

The Bottom Line:

This problem tests whether students understand the geometric meaning of "infinitely many solutions" (equations represent the same line) and can translate that insight into algebraic proportionality relationships. The fraction arithmetic, while tedious, is secondary to the conceptual breakthrough.