24x + y = 48 6x + y = 72 The solution to the given system of equations is \((\mathrm{x},...

1. TRANSLATE the problem information

• Given system of equations:
  • \(\mathrm{24x + y = 48}\) (Equation 1)
  • \(\mathrm{6x + y = 72}\) (Equation 2)
• Find: the value of y

2. INFER the most efficient solution strategy

• Notice both equations have the same y-term (+y)
• This means elimination by subtraction will directly eliminate y and solve for x
• Once we have x, we can substitute back to find y

3. SIMPLIFY using elimination method

• Subtract Equation 2 from Equation 1:
  \(\mathrm{(24x + y) - (6x + y) = 48 - 72}\)
• Distribute the negative:
  \(\mathrm{24x + y - 6x - y = -24}\)
• Combine like terms:
  \(\mathrm{18x = -24}\)
• Solve for x:
  \(\mathrm{x = \frac{-24}{18} = \frac{-4}{3}}\)

4. SIMPLIFY by substitution to find y

• Substitute \(\mathrm{x = \frac{-4}{3}}\) into Equation 2:
  \(\mathrm{6(\frac{-4}{3}) + y = 72}\)
• Calculate:
  \(\mathrm{-8 + y = 72}\)
• Solve for y:
  \(\mathrm{y = 80}\)

Answer: 80

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when subtracting equations or during arithmetic calculations.

For example, they might incorrectly compute \(\mathrm{48 - 72 = 24}\) instead of \(\mathrm{-24}\), leading to \(\mathrm{x = \frac{24}{18} = \frac{4}{3}}\). When they substitute this back, they get \(\mathrm{6(\frac{4}{3}) + y = 72}\), so \(\mathrm{8 + y = 72}\), giving \(\mathrm{y = 64}\). This leads to confusion as none of the standard multiple choice options would match this incorrect result.

Second Most Common Error:

Poor INFER reasoning: Students don't recognize the elimination opportunity and instead try to solve using substitution from the start.

They might solve the first equation for y: \(\mathrm{y = 48 - 24x}\), then substitute into the second equation: \(\mathrm{6x + (48 - 24x) = 72}\). This gives \(\mathrm{6x + 48 - 24x = 72}\), so \(\mathrm{-18x = 24}\), leading to \(\mathrm{x = \frac{-4}{3}}\) (correct), but the extra algebraic steps create more opportunities for computational errors along the way.

The Bottom Line:

This problem rewards students who can quickly spot the elimination opportunity and execute clean arithmetic. The key insight is recognizing that identical y-coefficients make subtraction the most direct path to the solution.