Question:The graph of a line in the xy-plane passes through the points \((1, 6)\) and \((3, -1)\). This line has...

1. TRANSLATE the problem information

• Given information:
  • Line passes through points \((1, 6)\) and \((3, -1)\)
  • Line has x-intercept at \((\mathrm{a}, 0)\) and y-intercept at \((0, \mathrm{b})\)
  • Need to find \(\mathrm{b/a}\)

2. INFER the solution strategy

• To find the ratio \(\mathrm{b/a}\), we first need to determine the values of a and b
• This requires finding the equation of the line, then using the definitions of intercepts
• Start by finding the line equation using the two given points

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m} = \frac{\mathrm{y_2 - y_1}}{\mathrm{x_2 - x_1}}\)
• \(\mathrm{m} = \frac{-1 - 6}{3 - 1} = \frac{-7}{2}\)

4. SIMPLIFY to find the line equation

• Using point-slope form with point \((1, 6)\):
  \(\mathrm{y - 6} = \left(\frac{-7}{2}\right)(\mathrm{x - 1})\)
• Distribute: \(\mathrm{y - 6} = \frac{-7}{2}\mathrm{x} + \frac{7}{2}\)
• Add 6 to both sides: \(\mathrm{y} = \frac{-7}{2}\mathrm{x} + \frac{7}{2} + 6\)
• Combine terms: \(\mathrm{y} = \frac{-7}{2}\mathrm{x} + \frac{19}{2}\)

5. INFER the y-intercept value

• From slope-intercept form \(\mathrm{y} = \mathrm{mx} + \mathrm{b}\), we can see \(\mathrm{b} = \frac{19}{2}\)
• This means the y-intercept is at point \(\left(0, \frac{19}{2}\right)\)

6. SIMPLIFY to find the x-intercept

• Set y = 0 in our equation: \(0 = \frac{-7}{2}\mathrm{x} + \frac{19}{2}\)
• Subtract \(\frac{19}{2}\): \(\frac{-19}{2} = \frac{-7}{2}\mathrm{x}\)
• Multiply both sides by \(\frac{-2}{7}\): \(\mathrm{x} = \frac{19}{2} \times \frac{2}{7} = \frac{19}{7}\)
• So \(\mathrm{a} = \frac{19}{7}\), meaning x-intercept is at \(\left(\frac{19}{7}, 0\right)\)

7. SIMPLIFY to calculate the final ratio

• \(\mathrm{b/a} = \frac{19}{2} \div \frac{19}{7} = \frac{19}{2} \times \frac{7}{19} = \frac{7}{2}\)

Answer: \(\frac{7}{2}\) (Choice D)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when calculating the slope or converting between equation forms. For example, they might calculate the slope as \(\frac{7}{2}\) instead of \(\frac{-7}{2}\), or make sign errors when distributing in the point-slope form.

If they get the wrong slope, their entire line equation becomes incorrect, leading to wrong intercept values. A slope of \(\frac{7}{2}\) would eventually give them \(\mathrm{b/a} = \frac{-7}{2}\).

This may lead them to select Choice A \(\left(\frac{-7}{2}\right)\).

Second Most Common Error:

Poor INFER reasoning about the solution sequence: Students might try to find intercepts directly from the given points without first establishing the complete line equation. They may confuse the given points with the actual intercepts, or attempt shortcuts that don't work.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem requires systematic equation-building before finding specific values. Students who rush to find intercepts without properly establishing the line equation often make critical errors early in their solution process.