In the xy-plane, line p has a slope of -{5/3} and an x-intercept of \((-6, 0)\). What is the y-coordinate...

1. TRANSLATE the problem information

• Given information:
  • Slope of line p: \(\mathrm{m = -\frac{5}{3}}\)
  • x-intercept of line p: \(\mathrm{(-6, 0)}\)
• What we need to find: The y-coordinate of the y-intercept (the value of b in \(\mathrm{y = mx + b}\))

2. INFER the solution strategy

• Since we know the slope and one point on the line, we can use the slope-intercept form \(\mathrm{y = mx + b}\)
• The x-intercept \(\mathrm{(-6, 0)}\) means when \(\mathrm{x = -6, y = 0}\)
• We can substitute this point into our equation to solve for b

3. Set up the equation with known values

• Start with: \(\mathrm{y = mx + b}\)
• Substitute \(\mathrm{m = -\frac{5}{3}}\): \(\mathrm{y = -\frac{5}{3}x + b}\)
• Substitute the x-intercept point \(\mathrm{(-6, 0)}\): \(\mathrm{0 = (-\frac{5}{3})(-6) + b}\)

4. SIMPLIFY to solve for b

• Calculate \(\mathrm{(-\frac{5}{3})(-6)}\):
  • The negatives cancel: \(\mathrm{(-\frac{5}{3})(-6) = (\frac{5}{3})(6)}\)
  • Multiply: \(\mathrm{\frac{5 \times 6}{3} = \frac{30}{3} = 10}\)
• So our equation becomes: \(\mathrm{0 = 10 + b}\)
• Solve for b: \(\mathrm{b = -10}\)

Answer: -10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse x-intercept with y-intercept concepts and may try to use \(\mathrm{(-6, 0)}\) as if it represents the y-intercept directly.

They might think: "The x-intercept is \(\mathrm{(-6, 0)}\), so the y-intercept must be related to -6 somehow" and incorrectly conclude the y-intercept is -6 or 6, rather than understanding that the x-intercept is just another point on the line that helps us find b.

This leads to confusion and guessing rather than systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when calculating \(\mathrm{(-\frac{5}{3})(-6)}\), either forgetting that two negatives make a positive or making arithmetic mistakes with fractions.

For example, they might calculate \(\mathrm{(-\frac{5}{3})(-6) = -10}\) instead of \(\mathrm{+10}\), leading them to solve \(\mathrm{0 = -10 + b}\) and get \(\mathrm{b = 10}\) instead of \(\mathrm{b = -10}\).

This may lead them to select an incorrect positive answer if available in multiple choice options.

The Bottom Line:

This problem requires students to distinguish between different types of intercepts and understand that any point on a line (including the x-intercept) can be used with the slope to find the complete equation. The key insight is recognizing that the x-intercept gives us a coordinate pair to substitute into the slope-intercept form.