The function \(\mathrm{k(x) = ax + 7}\), where a is a constant. If \(\mathrm{k(3) - k(1) = 10}\), what is...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{k(x) = ax + 7}\) (linear function with unknown coefficient a)
  • \(\mathrm{k(3) - k(1) = 10}\) (condition relating two function values)
• What this tells us: We need to find the specific value of coefficient a

2. TRANSLATE the function values

• Evaluate each function value by substituting:
  • \(\mathrm{k(3) = a(3) + 7 = 3a + 7}\)
  • \(\mathrm{k(1) = a(1) + 7 = a + 7}\)

3. TRANSLATE the given condition into an equation

• Set up the equation: \(\mathrm{k(3) - k(1) = 10}\)
• Substitute our expressions: \(\mathrm{(3a + 7) - (a + 7) = 10}\)

4. SIMPLIFY the equation algebraically

• Distribute the negative sign carefully: \(\mathrm{3a + 7 - a - 7 = 10}\)
• Combine like terms: \(\mathrm{2a = 10}\)
• Solve for a: \(\mathrm{a = 5}\)

Answer: C (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making sign errors when subtracting \(\mathrm{(a + 7)}\)

Students often write: \(\mathrm{(3a + 7) - (a + 7) = 3a + 7 - a + 7 = 2a + 14}\)

This gives them \(\mathrm{2a + 14 = 10}\), so \(\mathrm{2a = -4}\), and \(\mathrm{a = -2}\). Since this isn't among the choices, they may get confused and select Choice A (2) by dropping the negative sign or making another computational error.

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding what \(\mathrm{k(3) - k(1) = 10}\) means

Some students might think this means k(3) should equal 10 by itself, leading them to solve \(\mathrm{3a + 7 = 10}\), which gives \(\mathrm{3a = 3}\), so \(\mathrm{a = 1}\). Since 1 isn't a choice, they might guess or select Choice B (3) as the coefficient they calculated.

The Bottom Line:

This problem tests your ability to work systematically with function notation and maintain accuracy through algebraic manipulation. The key is careful attention to signs when subtracting expressions.