6 + 7r = pw 7r - 5w = 5w + 11 In the given system of equations, p is...

1. TRANSLATE the problem information

• Given system of equations:
  • \(\mathrm{6 + 7r = pw}\)
  • \(\mathrm{7r - 5w = 5w + 11}\)
• Need to find: value of p that makes the system have no solution

2. INFER the approach

• A system has no solution when the equations are inconsistent (lead to a contradiction)
• Strategy: Solve both equations for the same expression, then set them equal

3. SIMPLIFY each equation to isolate 7r

• From first equation: \(\mathrm{6 + 7r = pw}\)
  Subtract 6: \(\mathrm{7r = pw - 6}\)

• From second equation: \(\mathrm{7r - 5w = 5w + 11}\)
  Add 5w to both sides: \(\mathrm{7r = 10w + 11}\)

4. INFER the consistency condition

• Since both expressions equal 7r, set them equal:
  \(\mathrm{pw - 6 = 10w + 11}\)

5. SIMPLIFY to find when contradiction occurs

• Rearrange: \(\mathrm{pw - 10w = 17}\)
• Factor: \(\mathrm{w(p - 10) = 17}\)

6. INFER the no-solution condition

• For no solution, we need an impossible equation like \(\mathrm{0 = 17}\)
• This happens when \(\mathrm{p - 10 = 0}\) (making the left side 0) but right side = 17
• Therefore: \(\mathrm{p = 10}\)

Answer: 10

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" means one of the variables can't be found, rather than recognizing it means the system is mathematically inconsistent. They might try to solve for specific values of r and w instead of looking for the contradiction condition. This leads to confusion and guessing rather than systematic analysis.

Second Most Common Error:

Inadequate SIMPLIFY execution: Making algebraic errors when rearranging equations

Students might incorrectly manipulate the second equation (\(\mathrm{7r - 5w = 5w + 11}\)) and get \(\mathrm{7r = 5w + 11}\) instead of \(\mathrm{7r = 10w + 11}\). This error in combining like terms leads to the wrong equation \(\mathrm{pw - 6 = 5w + 11}\), giving a different value for p.

The Bottom Line:

This problem requires understanding the abstract concept of when systems are inconsistent, not just mechanical equation-solving skills. Students must recognize that \(\mathrm{p = 10}\) creates a mathematical impossibility (\(\mathrm{0 = 17}\)), which is exactly what "no solution" means.