Question:10x - y = -1154x + y = 94The solution to the given system of equations is \((\mathrm{x}, \mathrm{y})\). What...

1. INFER the best solution strategy

• Given system:
  • \(\mathrm{10x - y = -115}\)
  • \(\mathrm{4x + y = 94}\)
• Key insight: The y coefficients are opposites (\(\mathrm{-1}\) and \(\mathrm{+1}\))
• This means adding the equations will eliminate y completely
• Choose elimination method over substitution

2. SIMPLIFY by adding the equations

• Add left sides and right sides:
  \(\mathrm{(10x - y) + (4x + y) = -115 + 94}\)
• The y terms cancel: \(\mathrm{-y + y = 0}\)
• Combine like terms: \(\mathrm{14x = -21}\)
• Solve for x: \(\mathrm{x = -21/14 = -3/2}\)

3. SIMPLIFY by substituting back

• Use the simpler second equation: \(\mathrm{4x + y = 94}\)
• Substitute \(\mathrm{x = -3/2}\):
  \(\mathrm{4(-3/2) + y = 94}\)
• Calculate: \(\mathrm{-6 + y = 94}\)
• Solve for y: \(\mathrm{y = 100}\)

Answer: D) 100

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Sign errors when adding equations or arithmetic mistakes with fractions

Students often make errors like:

• Incorrectly adding: \(\mathrm{10x + 4x = 15x}\) (instead of 14x)
• Sign errors: \(\mathrm{-115 + 94 = 21}\) (instead of -21)
• Substitution errors: \(\mathrm{4(-3/2) = 6}\) (forgetting the negative)

These arithmetic mistakes lead to incorrect values of x, which then produce wrong y values, causing them to select incorrect answer choices.

Second Most Common Error:

Poor INFER reasoning: Not recognizing the elimination opportunity and choosing substitution instead

Some students don't notice that the y coefficients are opposites and attempt substitution from the start. While this can work, it involves more complex fraction arithmetic and increases chances for errors. They might solve the first equation for y: \(\mathrm{y = 10x + 115}\), then substitute into the second equation, leading to messier calculations.

The Bottom Line:

This problem rewards students who can spot the strategic advantage of elimination and execute clean arithmetic with signed numbers and fractions.