In the xy-plane, line t passes through the points \((0, 9)\) and \((1, 17)\). Which equation defines line t?

1. TRANSLATE the problem information

• Given information:
  • Line t passes through points \(\mathrm{(0, 9)}\) and \(\mathrm{(1, 17)}\)
  • Need to find equation in the form \(\mathrm{y = mx + b}\)

• What this tells us: We need to find both the slope (m) and y-intercept (b)

2. INFER the most efficient approach

• Since we have the point \(\mathrm{(0, 9)}\), we can directly identify the y-intercept
• When \(\mathrm{x = 0, y = 9}\), so \(\mathrm{b = 9}\)
• We'll use the slope formula with both points to find m

3. SIMPLIFY the slope calculation

• Apply slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• Using \(\mathrm{(0, 9)}\) as \(\mathrm{(x_1, y_1)}\) and \(\mathrm{(1, 17)}\) as \(\mathrm{(x_2, y_2)}\):
  \(\mathrm{m = \frac{17 - 9}{1 - 0}}\)
  \(\mathrm{= \frac{8}{1}}\)
  \(\mathrm{= 8}\)

4. INFER the final equation

• Substitute \(\mathrm{m = 8}\) and \(\mathrm{b = 9}\) into \(\mathrm{y = mx + b}\)
• Result: \(\mathrm{y = 8x + 9}\)

Answer: D. \(\mathrm{y = 8x + 9}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not recognize that the point \(\mathrm{(0, 9)}\) directly gives them the y-intercept. Instead, they might try to use both points in some complex substitution process, leading to confusion about which values represent what parts of the equation.

This leads to abandoning systematic solution and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the slope formula but make arithmetic errors, such as calculating \(\mathrm{\frac{17-9}{1-0}}\) as \(\mathrm{\frac{8}{8} = 1}\) instead of \(\mathrm{\frac{8}{1} = 8}\), or reversing the fraction to get \(\mathrm{\frac{1}{8}}\).

This may lead them to select Choice A (\(\mathrm{y = \frac{1}{8}x + 9}\)) or Choice C (\(\mathrm{y = x + 8}\)).

The Bottom Line:

This problem tests whether students can efficiently use coordinate information - recognizing shortcuts (like reading y-intercept directly) while systematically applying the slope formula. The key insight is that having the y-intercept point \(\mathrm{(0, 9)}\) eliminates half the work.