5y = 10x + 11-{5y = 5x - 21}The solution to the given system of equations is \((\mathrm{x}, \mathrm{y})\). What...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{5y = 10x + 11}\)
  • Second equation: \(\mathrm{-5y = 5x - 21}\)
  • Need to find: the value of \(\mathrm{30x}\)

2. INFER the solution strategy

• Notice that the y-coefficients are opposites: \(\mathrm{+5y}\) and \(\mathrm{-5y}\)
• This means we can add the equations together to eliminate y completely
• This is more efficient than substitution for this particular system

3. SIMPLIFY by adding the equations

• Add left sides: \(\mathrm{5y + (-5y) = 0}\)
• Add right sides: \(\mathrm{(10x + 11) + (5x - 21) = 15x - 10}\)
• Combined equation: \(\mathrm{0 = 15x - 10}\)

4. SIMPLIFY to solve for the x relationship

• From \(\mathrm{0 = 15x - 10}\), add 10 to both sides: \(\mathrm{10 = 15x}\)
• We could solve for x, but we need \(\mathrm{30x}\)
• Multiply both sides by 2: \(\mathrm{20 = 30x}\)

Answer: 20

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that the coefficients of y are opposites, so they attempt substitution instead of the more efficient elimination method. This leads to unnecessarily complex algebra with fractions, increasing the chance of arithmetic errors. This approach can still work but is much more prone to calculation mistakes that could lead to an incorrect final answer.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the elimination strategy but make arithmetic errors when combining like terms. For example, they might get \(\mathrm{15x + 10 = 0}\) instead of \(\mathrm{15x - 10 = 0}\), leading to \(\mathrm{x = -\frac{2}{3}}\) instead of \(\mathrm{x = \frac{2}{3}}\). This would give \(\mathrm{30x = -20}\), which doesn't match any reasonable answer choice and causes confusion.

The Bottom Line:

This problem rewards students who can quickly identify that elimination is the optimal strategy and execute the algebra cleanly. The key insight is recognizing that when coefficients are exact opposites, addition eliminates that variable immediately.