The graph of 7x + 2y = -31 in the xy-plane has an x-intercept at \((\mathrm{a}, 0)\) and a y-intercept...

1. TRANSLATE the intercept definitions into algebraic work

• Given information:
  • Linear equation: \(\mathrm{7x + 2y = -31}\)
  • x-intercept is at point \(\mathrm{(a, 0)}\)
  • y-intercept is at point \(\mathrm{(0, b)}\)

• What this tells us:
  • At x-intercept: \(\mathrm{y = 0}\), so we substitute \(\mathrm{y = 0}\) into our equation
  • At y-intercept: \(\mathrm{x = 0}\), so we substitute \(\mathrm{x = 0}\) into our equation

2. SIMPLIFY to find the x-intercept value a

• Substitute \(\mathrm{y = 0}\) into \(\mathrm{7x + 2y = -31}\):
  \(\mathrm{7x + 2(0) = -31}\)
  \(\mathrm{7x = -31}\)
  \(\mathrm{x = -31/7}\)

• Therefore: \(\mathrm{a = -31/7}\)

3. SIMPLIFY to find the y-intercept value b

• Substitute \(\mathrm{x = 0}\) into \(\mathrm{7x + 2y = -31}\):
  \(\mathrm{7(0) + 2y = -31}\)
  \(\mathrm{2y = -31}\)
  \(\mathrm{y = -31/2}\)

• Therefore: \(\mathrm{b = -31/2}\)

4. SIMPLIFY the fraction division to find b/a

• Set up the division: \(\mathrm{b/a = (-31/2) ÷ (-31/7)}\)

• Convert to multiplication: \(\mathrm{(-31/2) × (7/-31)}\)

• The -31 terms cancel: \(\mathrm{(-31/2) × (7/-31) = 7/2}\)

Answer: D. 7/2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which coordinate equals zero for each intercept type. They might substitute \(\mathrm{x = 0}\) to find the x-intercept or \(\mathrm{y = 0}\) to find the y-intercept (backwards from correct approach).

If they find x-intercept by setting \(\mathrm{x = 0}\), they get \(\mathrm{y = -31/2}\), making them think \(\mathrm{a = -31/2}\).
If they find y-intercept by setting \(\mathrm{y = 0}\), they get \(\mathrm{x = -31/7}\), making them think \(\mathrm{b = -31/7}\).

This gives them \(\mathrm{b/a = (-31/7) ÷ (-31/2) = (-31/7) × (2/-31) = 2/7}\), leading them to select Choice C (2/7).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly find \(\mathrm{a = -31/7}\) and \(\mathrm{b = -31/2}\) but make sign errors when dividing fractions with negatives.

They might compute \(\mathrm{b/a = (-31/2) ÷ (-31/7)}\) but forget that dividing two negatives gives a positive result, or they might incorrectly handle the fraction inversion step.

This could lead them to select Choice A (-7/2) or Choice B (-2/7) depending on where the sign error occurs.

The Bottom Line:

This problem tests whether students truly understand what intercepts mean algebraically (which variable to set to zero) and whether they can execute fraction division accurately with negative numbers.