Question:3sqrt(a + 1) - 2sqrt(b - 5) = 133sqrt(a + 1) + 2sqrt(b - 5) = 35The pair (a, b)...

1. TRANSLATE the problem information

• Given system:
  • \(3\sqrt{\mathrm{a + 1}} - 2\sqrt{\mathrm{b - 5}} = 13\)
  • \(3\sqrt{\mathrm{a + 1}} + 2\sqrt{\mathrm{b - 5}} = 35\)
• Find: value of a
• Constraints: \(\mathrm{a} \geq -1\) and \(\mathrm{b} \geq 5\)

2. INFER the strategic approach

• The radical expressions make this system look complicated, but notice both equations contain the same two radical terms: \(3\sqrt{\mathrm{a + 1}}\) and \(2\sqrt{\mathrm{b - 5}}\)
• Key insight: Use substitution to treat these complex expressions as simple variables
• This transforms a radical system into a basic linear system

3. SIMPLIFY using substitution

• Let \(\mathrm{U} = 3\sqrt{\mathrm{a + 1}}\) and \(\mathrm{V} = 2\sqrt{\mathrm{b - 5}}\)
• The system becomes:
  • \(\mathrm{U - V} = 13\)
  • \(\mathrm{U + V} = 35\)

4. SIMPLIFY the linear system

• Add both equations to eliminate V:
  \(\mathrm{(U - V) + (U + V)} = 13 + 35\)
  \(2\mathrm{U} = 48\)
  \(\mathrm{U} = 24\)

5. SIMPLIFY to find the original variable

• Since \(\mathrm{U} = 3\sqrt{\mathrm{a + 1}} = 24\):
  \(\sqrt{\mathrm{a + 1}} = 8\)
• Square both sides:
  \(\mathrm{a + 1} = 64\)
  \(\mathrm{a} = 63\)

Answer: 63

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to solve the system directly with the radical expressions rather than recognizing the substitution strategy.

They might try to isolate one radical in each equation, leading to expressions like \(\sqrt{\mathrm{a + 1}} = \frac{13 + 2\sqrt{\mathrm{b - 5}}}{3}\), creating an even more complex system. This approach becomes algebraically unwieldy and often leads to computational errors or complete confusion, causing them to abandon systematic solution and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the substitution approach but make arithmetic errors in the linear system solution.

For example, when adding \(\mathrm{U - V} = 13\) and \(\mathrm{U + V} = 35\), they might incorrectly get \(2\mathrm{U} = 22\) instead of \(2\mathrm{U} = 48\), leading to \(\mathrm{U} = 11\). This gives \(\sqrt{\mathrm{a + 1}} = \frac{11}{3}\), and ultimately \(\mathrm{a} \approx 12.4\), which doesn't match the expected integer answer and leads to confusion.

The Bottom Line:

This problem tests whether students can recognize when a strategic substitution dramatically simplifies what initially appears to be a complex system. The key insight is treating the radical expressions as single units rather than trying to manipulate them individually.