The y-intercept of the graph of 12x + 2y = 18 in the xy-plane is \((0, \mathrm{y})\). What is the...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(12\mathrm{x} + 2\mathrm{y} = 18\)
  • Need to find the y-intercept \((0, \mathrm{y})\)
• What this tells us: The y-intercept occurs where the line crosses the y-axis, which means \(\mathrm{x} = 0\)

2. SIMPLIFY by substitution and solving

• Substitute \(\mathrm{x} = 0\) into the equation: \(12\mathrm{x} + 2\mathrm{y} = 18\)
  • \(12(0) + 2\mathrm{y} = 18\)
  • \(0 + 2\mathrm{y} = 18\)
  • \(2\mathrm{y} = 18\)
• Solve for y: \(\mathrm{y} = 18 \div 2 = 9\)

Answer: 9

Why Students Usually Falter on This Problem

Most Common Error Path:

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

Students might confuse y-intercept with x-intercept and set \(\mathrm{y} = 0\) instead of \(\mathrm{x} = 0\). This leads them to solve:

\(12\mathrm{x} + 2(0) = 18\)
\(12\mathrm{x} = 18\)
\(\mathrm{x} = 1.5\)

This causes confusion since they're looking for a y-value but calculated an x-value, leading to guessing or abandoning the systematic solution.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors in basic division

Students correctly set up \(2\mathrm{y} = 18\) but make calculation errors, such as:

• \(\mathrm{y} = 18 \div 2 = 6\) (simple arithmetic mistake)
• Forgetting to divide both sides by 2, leaving \(\mathrm{y} = 18\)

These arithmetic slip-ups lead to selecting incorrect answer choices.

The Bottom Line:

This problem tests whether students truly understand coordinate geometry vocabulary. The mathematical operations are straightforward once students recognize that y-intercept means "where x equals zero."