In the coordinate plane, a point \(\mathrm{(x, y)}\) lies on both the lines y = -2x + 19 and x...

1. TRANSLATE the problem information

• Given information:
  • Point (x, y) lies on line \(\mathrm{y = -2x + 19}\)
  • Same point lies on line \(\mathrm{x = 6}\)
  • Need to find \(\mathrm{x + y}\)

• What this tells us: The point's coordinates must satisfy both equations simultaneously

2. INFER the solution strategy

• Since we have \(\mathrm{x = 6}\), we know the x-coordinate directly
• We can substitute this x-value into the other equation to find y
• Then we can calculate \(\mathrm{x + y}\)

3. SIMPLIFY to find the y-coordinate

• Substitute \(\mathrm{x = 6}\) into \(\mathrm{y = -2x + 19}\):
  \(\mathrm{y = -2(6) + 19}\)
  \(\mathrm{y = -12 + 19}\)
  \(\mathrm{y = 7}\)

4. Calculate the final answer

• \(\mathrm{x + y = 6 + 7 = 13}\)

Answer: C (13)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not realize that 'lies on both lines' means the coordinates satisfy both equations. They might try to solve the system algebraically by setting the equations equal to each other, getting confused because they can't set \(\mathrm{y = -2x + 19}\) equal to \(\mathrm{x = 6}\).

This leads to confusion and guessing, or they might incorrectly try to solve \(\mathrm{-2x + 19 = 6}\), getting \(\mathrm{x = 6.5}\), then substitute back incorrectly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that \(\mathrm{x = 6}\) and need to substitute, but make arithmetic errors in calculating \(\mathrm{y = -2(6) + 19}\). Common mistakes include:

• Forgetting the negative sign: \(\mathrm{y = 2(6) + 19 = 31}\), leading to \(\mathrm{x + y = 37}\) (not among choices)
• Sign errors in addition: \(\mathrm{y = -12 + 19 = -7}\), leading to \(\mathrm{x + y = -1}\) (not among choices)

This typically leads to confusion and guessing among the available choices.

The Bottom Line:

This problem tests whether students understand that intersection points satisfy all given equations simultaneously, and whether they can recognize when direct substitution is the most efficient approach rather than formal algebraic methods.