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 mathematical procedures

• Given information:
  • Linear equation: \(7\mathrm{x} + 2\mathrm{y} = -31\)
  • x-intercept at \((\mathrm{a}, 0)\) means when \(\mathrm{y} = 0\), \(\mathrm{x} = \mathrm{a}\)
  • y-intercept at \((0, \mathrm{b})\) means when \(\mathrm{x} = 0\), \(\mathrm{y} = \mathrm{b}\)
  • Need to find \(\mathrm{b}/\mathrm{a}\)
• What this tells us: To find intercepts, substitute 0 for one variable and solve for the other

2. Find the x-intercept by SIMPLIFYing when \(\mathrm{y} = 0\)

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

3. Find the y-intercept by SIMPLIFYing when \(\mathrm{x} = 0\)

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

4. SIMPLIFY the fraction division \(\mathrm{b}/\mathrm{a}\)

• Calculate: \(\mathrm{b}/\mathrm{a} = (-31/2) \div (-31/7)\)
• Convert division to multiplication: \((-31/2) \times (-7/31)\)
• The negatives cancel: \((31/2) \times (7/31)\)
• The 31s cancel: \((1/2) \times 7 = 7/2\)

Answer: D. \(7/2\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Sign errors during fraction division

Students correctly find \(\mathrm{a} = -31/7\) and \(\mathrm{b} = -31/2\), but make mistakes when calculating \(\mathrm{b}/\mathrm{a} = (-31/2) \div (-31/7)\). They might:

• Forget that dividing two negatives gives a positive result
• Incorrectly flip the wrong fraction
• Make arithmetic errors when canceling terms

This may lead them to select Choice A (\(-7/2\)) or Choice B (\(-2/7\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Confusing which variable to set to zero for each intercept

Students might substitute \(\mathrm{x} = 0\) to find the x-intercept and \(\mathrm{y} = 0\) to find the y-intercept (backwards from correct). This leads to finding the intercepts at the wrong coordinates and calculating \(\mathrm{a}/\mathrm{b}\) instead of \(\mathrm{b}/\mathrm{a}\).

This may lead them to select Choice C (\(2/7\)).

The Bottom Line:

The key challenge is correctly implementing the intercept definitions and carefully handling negative fraction arithmetic. Students who remember that "x-intercept means \(\mathrm{y} = 0\)" and "y-intercept means \(\mathrm{x} = 0\)" while staying organized with negative signs will succeed.