8x + y = 5 y = 9x + 1 The solution to the given system of equations is \((\mathrm{x},...

1. TRANSLATE the problem information

• Given system:
  • \(8\mathrm{x} + \mathrm{y} = 5\)
  • \(\mathrm{y} = 9\mathrm{x} + 1\)
• We need to find the value of x

2. INFER the solution strategy

• Notice that the second equation already has y isolated: \(\mathrm{y} = 9\mathrm{x} + 1\)
• This makes substitution the most efficient method
• We can substitute the expression for y into the first equation

3. SIMPLIFY by substituting and solving

• Substitute \(\mathrm{y} = 9\mathrm{x} + 1\) into the first equation:
  \(8\mathrm{x} + (9\mathrm{x} + 1) = 5\)
• Combine like terms:
  \(8\mathrm{x} + 9\mathrm{x} + 1 = 5\)
  \(17\mathrm{x} + 1 = 5\)
• Isolate x:
  \(17\mathrm{x} = 5 - 1\)
  \(17\mathrm{x} = 4\)
  \(\mathrm{x} = \frac{4}{17}\)

Answer: \(\frac{4}{17}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic mistakes when combining like terms or solving for x.

Common mistakes include:

• Combining 8x + 9x incorrectly (getting 16x or 18x instead of 17x)
• Sign errors when moving the constant: writing 17x = 5 + 1 = 6 instead of 17x = 5 - 1 = 4
• Division errors when computing 4/17

These calculation errors can lead them to select Choice A (-6) or Choice D (4) instead of the correct fractional answer.

Second Most Common Error:

Poor INFER reasoning: Students might choose elimination method instead of the more straightforward substitution, making the problem unnecessarily complex and increasing chances for errors.

They might try to eliminate variables by multiplying equations, creating messier arithmetic that leads to mistakes and confusion about the final answer.

The Bottom Line:

This problem tests whether students can recognize when substitution is the clearest path forward and then execute basic algebraic manipulation accurately. The fractional answer requires careful arithmetic attention.