In the given equation, \(\frac{12\mathrm{x}+28}{4} - \frac{\mathrm{s}}{13} = \mathrm{r}(\mathrm{x} - 8)\), s and r are constants, and s gt 0....

1. TRANSLATE the problem information

• Given equation: \(\frac{12x+28}{4} - \frac{s}{13} = r(x - 8)\)
• \(s\) and \(r\) are constants, \(s \gt 0\)
• The equation has infinitely many solutions
• Find: value of \(s\)

2. INFER the key mathematical condition

• For a linear equation to have infinitely many solutions, both sides must be identical after simplification
• This means: coefficients of \(x\) must be equal AND constant terms must be equal

3. SIMPLIFY the equation to standard form

• Left side: \(\frac{12x+28}{4} - \frac{s}{13} = 3x + 7 - \frac{s}{13}\)
• Right side: \(r(x - 8) = rx - 8r\)
• Rewritten equation: \(3x + 7 - \frac{s}{13} = rx - 8r\)

4. INFER and apply the matching conditions

• For coefficients of \(x\): \(3 = r\)
• Therefore: \(r = 3\)

5. SIMPLIFY to find s using the constant terms condition

• Constants must be equal: \(7 - \frac{s}{13} = -8r\)
• Substitute \(r = 3\): \(7 - \frac{s}{13} = -8(3) = -24\)
• Solve: \(7 - \frac{s}{13} = -24\)
• Add \(\frac{s}{13}\) to both sides: \(7 = -24 + \frac{s}{13}\)
• Add 24 to both sides: \(31 = \frac{s}{13}\)
• Multiply by 13: \(s = 31 \times 13 = 403\)

Answer: 403

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize what "infinitely many solutions" means mathematically. They might try to solve the equation for \(x\) instead of understanding that infinitely many solutions requires the equation to be an identity (both sides identical). This leads to confusion about what to do with the given information, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that coefficients and constants must match, but make arithmetic errors when solving \(7 - \frac{s}{13} = -24\). Common mistakes include sign errors or incorrect fraction operations. For example, incorrectly getting \(\frac{s}{13} = -17\) instead of \(\frac{s}{13} = 31\), which would lead to \(s = -221\) instead of 403.

The Bottom Line:

This problem tests understanding of what makes an equation have infinitely many solutions - a concept that requires students to think beyond just "solving for \(x\)" and recognize when an equation becomes an identity.