Question:A line in the xy-plane is defined by the equation 5x + 8y = 120. What is the x-intercept of...

1. TRANSLATE the problem information

• Given information:
  • Line equation: \(\mathrm{5x + 8y = 120}\)
  • Need to find: x-intercept

• What this tells us: The x-intercept is where the line crosses the x-axis, so the y-coordinate must equal 0

2. TRANSLATE the intercept concept into mathematical form

• At the x-intercept: \(\mathrm{y = 0}\)
• Substitute this into our equation: \(\mathrm{5x + 8(0) = 120}\)

3. SIMPLIFY the equation

• \(\mathrm{5x + 8(0) = 120}\)
• \(\mathrm{5x + 0 = 120}\)
• \(\mathrm{5x = 120}\)
• \(\mathrm{x = 120 ÷ 5 = 24}\)

4. Express the final answer as a coordinate point

• Since \(\mathrm{y = 0}\) and \(\mathrm{x = 24}\), the x-intercept is \(\mathrm{(24, 0)}\)

Answer: (D) \(\mathrm{(24, 0)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse x-intercept with y-intercept and set \(\mathrm{x = 0}\) instead of \(\mathrm{y = 0}\).

They substitute \(\mathrm{x = 0}\) into the equation: \(\mathrm{5(0) + 8y = 120}\), which gives \(\mathrm{8y = 120}\), so \(\mathrm{y = 15}\). They might then incorrectly write this as \(\mathrm{(15, 0)}\) instead of \(\mathrm{(0, 15)}\).

This may lead them to select Choice (C) \(\mathrm{(15, 0)}\).

The Bottom Line:

The key challenge is correctly translating "x-intercept" into the mathematical condition \(\mathrm{y = 0}\). Once this translation is made, the algebra is straightforward. Students who master this concept of intercepts will find these problems much more manageable.