The line in the xy-plane passes through the points \((2, 7)\) and \((6, -7)\). This line has an x-intercept at...

1. TRANSLATE the problem information

• Given information:
  • Line passes through points \((2, 7)\) and \((6, -7)\)
  • Line has x-intercept at \((\mathrm{a}, 0)\) and y-intercept at \((0, \mathrm{b})\)
  • Need to find \(\frac{\mathrm{b}}{\mathrm{a}}\)

2. INFER the solution approach

• To find intercepts, we first need the equation of the line
• We can find the line equation using the slope formula and point-slope form
• Once we have the equation, we can find where it crosses each axis

3. SIMPLIFY to find the slope

Using the slope formula with our two points:

\(\mathrm{m} = \frac{-7 - 7}{6 - 2}\)
\(\mathrm{m} = \frac{-14}{4}\)
\(\mathrm{m} = -\frac{7}{2}\)

4. SIMPLIFY to find the line equation

Using point-slope form with \((2, 7)\):

\(\mathrm{y - 7} = \left(-\frac{7}{2}\right)(\mathrm{x - 2})\)
\(\mathrm{y - 7} = -\frac{7}{2}\mathrm{x} + 7\)
\(\mathrm{y} = -\frac{7}{2}\mathrm{x} + 14\)

5. INFER how to find each intercept

• For x-intercept: set \(\mathrm{y} = 0\) and solve for \(\mathrm{x}\)
• For y-intercept: the constant term gives us \(\mathrm{b}\) directly

6. SIMPLIFY to find the x-intercept (value of a)

Setting \(\mathrm{y} = 0\):

\(0 = -\frac{7}{2}\mathrm{x} + 14\)
\(\frac{7}{2}\mathrm{x} = 14\)
\(\mathrm{x} = 14 \times \frac{2}{7} = 4\)
Therefore, \(\mathrm{a} = 4\)

7. TRANSLATE to find the y-intercept (value of b)

From \(\mathrm{y} = -\frac{7}{2}\mathrm{x} + 14\), we see that \(\mathrm{b} = 14\)

8. SIMPLIFY to find the final ratio

\(\frac{\mathrm{b}}{\mathrm{a}} = \frac{14}{4}\)
\(\frac{\mathrm{b}}{\mathrm{a}} = \frac{7}{2}\)

Answer: 7/2 or Choice (D)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may try to work directly with intercept definitions without first finding the line equation. They might attempt to use the given points as if they were the intercepts themselves, or try to find intercepts through geometric reasoning rather than algebraic methods.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when converting from point-slope to slope-intercept form, or when solving \(0 = -\frac{7}{2}\mathrm{x} + 14\) for the x-intercept. Common mistakes include sign errors or fraction arithmetic mistakes.

This may lead them to select Choice (A) \(-\frac{7}{2}\) if they incorrectly calculate the ratio as slope instead of \(\frac{\mathrm{b}}{\mathrm{a}}\), or other incorrect choices based on their computational errors.

The Bottom Line:

This problem requires students to see that intercept problems are really line equation problems in disguise. The key insight is recognizing that you must establish the line equation first, then use it systematically to find where the line crosses each axis.