A line passes through the points \((4, 6)\) and \((15, 24)\) in the xy-plane. What is the slope of the...

1. INFER what the problem is asking

• We need to find the slope of a line passing through two specific points
• This requires using the slope formula to calculate the rate of change between the points

2. TRANSLATE the given information

• Point 1: \((4, 6)\) → \(\mathrm{x_1 = 4}\), \(\mathrm{y_1 = 6}\)
• Point 2: \((15, 24)\) → \(\mathrm{x_2 = 15}\), \(\mathrm{y_2 = 24}\)
• Apply slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)

3. TRANSLATE by substituting values into the formula

• \(\mathrm{m = \frac{24 - 6}{15 - 4}}\)
• Be careful with the order: y-values on top, x-values on bottom

4. SIMPLIFY the arithmetic

• \(\mathrm{m = \frac{18}{11}}\)
• This fraction is already in simplest form since 18 and 11 share no common factors

Answer: \(\mathrm{\frac{18}{11}}\) (or 1.636 as a decimal)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up which coordinate is x and which is y, or they reverse the order in the slope formula.

For example, they might use \((6, 4)\) and \((24, 15)\) instead of the correct coordinates, or they might calculate \(\mathrm{\frac{x_2 - x_1}{y_2 - y_1}}\) instead of \(\mathrm{\frac{y_2 - y_1}{x_2 - x_1}}\). This leads to completely incorrect numerical answers and causes confusion when trying to match their result to the expected answer format.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make arithmetic errors in the subtraction steps.

They might calculate \(\mathrm{(24 - 6) = 17}\) instead of 18, or \(\mathrm{(15 - 4) = 12}\) instead of 11. These calculation mistakes lead to incorrect fractions that don't simplify to the correct answer, causing them to doubt their approach even when their method is correct.

The Bottom Line:

This problem tests whether students can systematically apply a memorized formula while maintaining accuracy in coordinate identification and basic arithmetic. Success depends on careful attention to the order and position of values in both the coordinate pairs and the slope formula.