The graph of the equation 5x - 3y = 30 is a line in the xy-plane. This line, along with...

1. TRANSLATE the problem setup

• Given information:
  • Linear equation: \(\mathrm{5x - 3y = 30}\)
  • Line forms triangle with x-axis and y-axis
  • Need to find area of this triangle

• What this tells us: The triangle's vertices are at the origin \(\mathrm{(0,0)}\) and the two intercepts

2. INFER the solution strategy

• To find triangle area, we need base and height
• The triangle's legs lie along the coordinate axes
• Base = distance from origin to x-intercept
• Height = distance from origin to y-intercept
• Need to find where the line crosses each axis

3. SIMPLIFY to find the x-intercept

• Set \(\mathrm{y = 0}\) in the equation:
  \(\mathrm{5x - 3(0) = 30}\)
  \(\mathrm{5x = 30}\)
  \(\mathrm{x = 6}\)
• X-intercept is at \(\mathrm{(6, 0)}\)
• Base length = \(\mathrm{|6| = 6}\)

4. SIMPLIFY to find the y-intercept

• Set \(\mathrm{x = 0}\) in the equation:
  \(\mathrm{5(0) - 3y = 30}\)
  \(\mathrm{-3y = 30}\)
  \(\mathrm{y = -10}\)
• Y-intercept is at \(\mathrm{(0, -10)}\)
• Height length = \(\mathrm{|-10| = 10}\)

5. INFER and calculate the final area

• Apply triangle area formula: \(\mathrm{Area = \frac{1}{2} \times base \times height}\)
• \(\mathrm{Area = \frac{1}{2} \times 6 \times 10 = 30}\)

Answer: B) 30

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students don't recognize that "triangle formed by line and axes" means finding intercepts and using them as base and height.

Instead, they might try to use the slope or attempt more complex geometric calculations. This leads to confusion and abandoning systematic solution, causing them to guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when solving for intercepts, especially with the y-intercept calculation where \(\mathrm{-3y = 30}\) gives \(\mathrm{y = -10}\).

Getting \(\mathrm{y = 10}\) instead of \(\mathrm{y = -10}\) would give height = 10 (still correct due to absolute value), but getting confused about the negative might lead them to use wrong values entirely. This could cause calculation errors leading to selecting Choice A (15) or Choice C (45).

The Bottom Line:

This problem tests whether students can connect the geometric concept of "triangle formed by line and axes" to the algebraic process of finding intercepts, then apply basic area calculations.