2a + 8b = 198 2a + 4b = 98 The solution to the given system of equations is \(\mathrm{(a,...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{2a + 8b = 198}\) (Equation 1)
  • \(\mathrm{2a + 4b = 98}\) (Equation 2)
• We need to find the value of \(\mathrm{b}\)

2. INFER the solution strategy

• Notice both equations have the same coefficient for variable \(\mathrm{a}\) (both have \(\mathrm{2a}\))
• This suggests using elimination method - we can subtract one equation from the other to eliminate \(\mathrm{a}\)
• Subtracting will leave us with an equation containing only \(\mathrm{b}\)

3. SIMPLIFY by eliminating variable a

• Subtract Equation 2 from Equation 1:
  \(\mathrm{(2a + 8b) - (2a + 4b) = 198 - 98}\)
• Distribute the negative sign:
  \(\mathrm{2a + 8b - 2a - 4b = 100}\)
• Combine like terms:
  \(\mathrm{(2a - 2a) + (8b - 4b) = 100}\)
  \(\mathrm{0 + 4b = 100}\)
  \(\mathrm{4b = 100}\)

4. SIMPLIFY to solve for b

• Divide both sides by 4:
  \(\mathrm{b = 100 ÷ 4 = 25}\)

Answer: 25

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize the elimination strategy and instead attempt substitution, which is much more complicated for this system. They might solve one equation for \(\mathrm{a}\) in terms of \(\mathrm{b}\), then substitute into the other equation, leading to more complex algebraic manipulation and higher chance of arithmetic errors.

This leads to confusion and potential abandonment of systematic solution, causing them to guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the elimination approach but make sign errors when subtracting the equations. They might incorrectly compute \(\mathrm{(8b - 4b)}\) as \(\mathrm{12b}\) instead of \(\mathrm{4b}\), or make errors in the arithmetic when subtracting 98 from 198.

This leads to an incorrect value for \(\mathrm{b}\) and wrong answer selection.

The Bottom Line:

This problem rewards recognizing the efficient elimination path - the identical coefficients of \(\mathrm{a}\) make subtraction the clear choice. Students who miss this insight often get bogged down in unnecessary complexity.