Line t in the xy-plane has a slope of -{8/3} and passes through the point \((9, 10)\). Which equation defines...

1. TRANSLATE the problem information

• Given information:
  • Slope of line t: \(-\frac{8}{3}\)
  • Point on line t: \((9, 10)\)
• We need to find the equation of line t

2. INFER the approach

• Use slope-intercept form: \(\mathrm{y = mx + b}\)
• We know \(\mathrm{m = -\frac{8}{3}}\), so we need to find \(\mathrm{b}\) (the y-intercept)
• We can find \(\mathrm{b}\) by substituting the known point \((9, 10)\) into our equation

3. Set up the equation with known slope

• Start with: \(\mathrm{y = mx + b}\)
• Substitute \(\mathrm{m = -\frac{8}{3}}\): \(\mathrm{y = -\frac{8}{3}x + b}\)

4. SIMPLIFY to find the y-intercept

• Substitute the point \((9, 10)\): \(10 = -\frac{8}{3}(9) + \mathrm{b}\)
• Calculate: \(-\frac{8}{3} \times 9 = -24\)
• So: \(10 = -24 + \mathrm{b}\)
• Solve for \(\mathrm{b}\): \(\mathrm{b = 10 + 24 = 34}\)

5. Write the final equation

• Substitute \(\mathrm{b = 34}\) back into \(\mathrm{y = -\frac{8}{3}x + b}\)
• Final equation: \(\mathrm{y = -\frac{8}{3}x + 34}\)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skills: Students make arithmetic errors when calculating \(-\frac{8}{3} \times 9\), often getting \(-8 \times 9 = -72\) but forgetting to divide by 3, or making sign errors.

A common mistake is calculating \(-\frac{8}{3} \times 9\) as \(-\frac{8}{27}\) instead of \(-24\). This leads to \(10 = -\frac{8}{27} + \mathrm{b}\), giving \(\mathrm{b = 10 + \frac{8}{27}}\), which doesn't match any answer choice. This causes confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse point-slope form with slope-intercept form, or try to use the point coordinates incorrectly in the equation.

Some students might think the equation is \(\mathrm{y = -\frac{8}{3} + 10}\) (mixing up the slope and point coordinates), which would lead them toward Choice C (\(\mathrm{y = -\frac{8}{3} + 10}\)) if such confusion persists.

The Bottom Line:

This problem tests whether students can systematically apply the slope-intercept form and perform accurate algebraic substitution. The key challenge is maintaining precision in arithmetic while working with fractions and negative numbers.