The graph of the equation 5x + 3y = -10 is a line in the xy-plane. The line has an...

1. TRANSLATE the problem information

• Given: Linear equation \(\mathrm{5x + 3y = -10}\)
• Need to find: x-intercept at \(\mathrm{(p, 0)}\) and y-intercept at \(\mathrm{(0, q)}\), then calculate \(\mathrm{q/p}\)
• What this tells us: We need to find where the line crosses each axis

2. INFER the approach for finding intercepts

• Key insight: At the x-intercept, \(\mathrm{y = 0}\) (the line crosses the x-axis)
• At the y-intercept, \(\mathrm{x = 0}\) (the line crosses the y-axis)
• Strategy: Substitute these zero values into the equation to solve for the intercepts

3. Find the x-intercept \(\mathrm{(p, 0)}\)

• Set \(\mathrm{y = 0}\) in the equation: \(\mathrm{5x + 3(0) = -10}\)
• SIMPLIFY: \(\mathrm{5x = -10}\), so \(\mathrm{x = -2}\)
• Therefore: \(\mathrm{p = -2}\)

4. Find the y-intercept \(\mathrm{(0, q)}\)

• Set \(\mathrm{x = 0}\) in the equation: \(\mathrm{5(0) + 3y = -10}\)
• SIMPLIFY: \(\mathrm{3y = -10}\), so \(\mathrm{y = -10/3}\)
• Therefore: \(\mathrm{q = -10/3}\)

5. SIMPLIFY to find \(\mathrm{q/p}\)

• \(\mathrm{q/p = (-10/3)/(-2)}\)
• Convert to multiplication: \(\mathrm{(-10/3) \times (-1/2)}\)
• Calculate: \(\mathrm{10/6 = 5/3}\)

Answer: D) \(\mathrm{5/3}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding what intercepts mean geometrically, leading to confusion about when to set variables to zero.

Students might try to solve the equation simultaneously or get confused about which coordinate should be zero for each intercept. This leads to setting up incorrect equations and getting wrong values for \(\mathrm{p}\) and \(\mathrm{q}\).

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making sign errors when working with negative numbers, especially in the final calculation of \(\mathrm{q/p}\).

Students correctly find \(\mathrm{p = -2}\) and \(\mathrm{q = -10/3}\), but then calculate \(\mathrm{q/p = (-10/3)/(-2)}\) incorrectly. Common errors include forgetting that dividing two negatives gives a positive result, or making arithmetic mistakes with the fraction division.

This may lead them to select Choice A (\(\mathrm{-5/3}\)) or Choice B (\(\mathrm{-3/5}\)) by getting the wrong sign or flipping the fraction incorrectly.

The Bottom Line:

This problem tests whether students truly understand what intercepts represent and can execute multi-step algebra with negative fractions. The key insight is recognizing that intercepts occur when one coordinate is zero, not solving some complex system of equations.

6. Question Type - Student Response

Acceptable answer forms: \(\mathrm{5/3}\), \(\mathrm{1.667}\) (rounded), \(\mathrm{1\frac{2}{3}}\)