In the xy-plane, the line given by \(3(\mathrm{y} - 5) = -2(\mathrm{x} + 9)\) is graphed. What is the x-intercept...

1. TRANSLATE the problem requirements

• Given information:
  • Line equation: \(3(\mathrm{y} - 5) = -2(\mathrm{x} + 9)\)
  • Need to find: x-intercept

• What this tells us: The x-intercept occurs where the line crosses the x-axis, meaning \(\mathrm{y} = 0\)

2. TRANSLATE the solution strategy

• To find x-intercept: substitute \(\mathrm{y} = 0\) into the equation and solve for x
• This gives us the x-coordinate where the line hits the x-axis

3. SIMPLIFY by substituting y = 0

• Start with: \(3(\mathrm{y} - 5) = -2(\mathrm{x} + 9)\)
• Substitute \(\mathrm{y} = 0\): \(3(0 - 5) = -2(\mathrm{x} + 9)\)
• Simplify the left side: \(3(-5) = -2(\mathrm{x} + 9)\)
• This becomes: \(-15 = -2(\mathrm{x} + 9)\)

4. SIMPLIFY by distributing and isolating

• Distribute the right side: \(-15 = -2\mathrm{x} - 18\)
• Add 18 to both sides: \(-15 + 18 = -2\mathrm{x}\)
• Simplify: \(3 = -2\mathrm{x}\)
• Divide both sides by −2: \(\mathrm{x} = -\frac{3}{2}\)

5. TRANSLATE the final answer

• The x-intercept occurs at \(\mathrm{x} = -\frac{3}{2}\)
• Written as a point: \((-\frac{3}{2}, 0)\)

Answer: B. \((-\frac{3}{2}, 0)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not understanding what "x-intercept" means

Students may not realize that finding the x-intercept requires setting \(\mathrm{y} = 0\). They might try other approaches or get confused about which variable to solve for. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Sign errors in the final division step

Students correctly work through the algebra to get \(3 = -2\mathrm{x}\), but when dividing both sides by −2, they forget to apply the negative sign properly. They calculate \(\mathrm{x} = \frac{3}{2}\) instead of \(\mathrm{x} = -\frac{3}{2}\).

This may lead them to select Choice D. \((\frac{3}{2}, 0)\)

The Bottom Line:

This problem tests whether students understand the geometric meaning of x-intercept and can execute multi-step algebraic manipulation without sign errors. The key insight is recognizing that x-intercepts occur when \(\mathrm{y} = 0\), then carefully tracking negative signs throughout the solution.