x = 3y = |x - 6| + 9The graphs of the equations in the given system of equations intersect...

1. TRANSLATE the problem information

• Given system of equations:
  • \(\mathrm{x = 3}\) (vertical line)
  • \(\mathrm{y = |x - 6| + 9}\) (absolute value function)
• We need to find where these graphs intersect (the y-coordinate)

2. INFER the solution strategy

• Since \(\mathrm{x = 3}\) from the first equation, substitute this value into the second equation
• This will give us the y-coordinate of the intersection point

3. SIMPLIFY the absolute value expression

• Substitute \(\mathrm{x = 3}\) into \(\mathrm{y = |x - 6| + 9}\):
  \(\mathrm{y = |3 - 6| + 9}\)
• Evaluate inside the absolute value first:
  \(\mathrm{y = |-3| + 9}\)
• Apply absolute value definition (\(\mathrm{|-3| = 3}\)):
  \(\mathrm{y = 3 + 9}\)
• Final calculation:
  \(\mathrm{y = 12}\)

Answer: D (12)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly evaluate \(\mathrm{|-3|}\) as -3 instead of 3

They think: "\(\mathrm{|-3| = -3}\) because there's a negative sign there"

Following this logic: \(\mathrm{y = |-3| + 9 = -3 + 9 = 6}\)

This leads them to select Choice B (6)

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "intersection point" means and try to solve both equations simultaneously rather than substituting

They might attempt to set \(\mathrm{|x - 6| + 9 = 3}\), thinking both expressions equal y, leading to confusion about which equation represents what.

This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students truly understand absolute value notation and can execute simple substitution correctly. The absolute value evaluation is the critical step where most errors occur.