The line 4x + ky = 20 has y-intercept \((0, 5)\). What is the value of k? 2 4 5...

1. TRANSLATE the y-intercept information

• Given information:
  • Linear equation: \(\mathrm{4x + ky = 20}\)
  • Y-intercept: \(\mathrm{(0, 5)}\)
• What this tells us: At the y-intercept, \(\mathrm{x = 0}\) and \(\mathrm{y = 5}\)

2. INFER the solution strategy

• Since we know specific coordinate values that satisfy the equation, we can substitute them in
• This will give us an equation with only k as the unknown, which we can solve

3. SIMPLIFY by substituting and solving

• Substitute \(\mathrm{x = 0}\) and \(\mathrm{y = 5}\) into \(\mathrm{4x + ky = 20}\):
  \(\mathrm{4(0) + k(5) = 20}\)
• Simplify: \(\mathrm{0 + 5k = 20}\)
• Solve: \(\mathrm{5k = 20}\), so \(\mathrm{k = 4}\)

Answer: B) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not fully understand what "y-intercept \(\mathrm{(0, 5)}\)" means mathematically. They might think it just means \(\mathrm{y = 5}\) without recognizing that x must equal 0 at this point. This incomplete translation prevents them from setting up the substitution correctly, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly substitute but make arithmetic errors, such as thinking \(\mathrm{5k = 20}\) means \(\mathrm{k = 5}\) (confusing the coefficient with the result) or \(\mathrm{k = 20}\) (forgetting to divide). This may lead them to select Choice C (5) or Choice E (20).

The Bottom Line:

This problem tests whether students truly understand what a y-intercept represents as a coordinate pair and can translate that understanding into actionable mathematical information. The algebra itself is straightforward once the substitution is set up correctly.