What is the solution (x, y) to the given system of equations?y = 4x - 9y = 19

1. INFER the most efficient approach

• Given information:
  • First equation: \(\mathrm{y = 4x - 9}\)
  • Second equation: \(\mathrm{y = 19}\)

• Key insight: The second equation directly tells us \(\mathrm{y = 19}\), so we can substitute this value into the first equation rather than using elimination

2. SIMPLIFY by substitution

• Substitute \(\mathrm{y = 19}\) into the first equation:
  \(\mathrm{19 = 4x - 9}\)

• Add 9 to both sides:
  \(\mathrm{19 + 9 = 4x}\)
  \(\mathrm{28 = 4x}\)

• Divide both sides by 4:
  \(\mathrm{x = 7}\)

3. Form the solution

• Since \(\mathrm{x = 7}\) and \(\mathrm{y = 19}\), the solution is \(\mathrm{(7, 19)}\)

Answer: B. (7, 19)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making arithmetic errors when solving \(\mathrm{19 = 4x - 9}\)

Students might incorrectly add or subtract when moving terms, or make division errors. For example, they might calculate \(\mathrm{19 - 9 = 12}\) instead of \(\mathrm{19 + 9 = 28}\), leading to \(\mathrm{12 = 4x}\) and \(\mathrm{x = 3}\). This doesn't match any answer choice directly, leading to confusion and guessing.

Second Most Common Error:

Conceptual confusion about coordinate pairs: Switching the x and y values in the final answer

Some students correctly find \(\mathrm{x = 7}\) and \(\mathrm{y = 19}\) but then mix up the coordinate order, thinking the solution could be \(\mathrm{(19, 7)}\) instead of \(\mathrm{(7, 19)}\). This may lead them to select Choice D (19, 7).

The Bottom Line:

This problem tests whether students can recognize when substitution is the most direct method and execute basic algebraic operations accurately. The key insight is immediately using the given y-value rather than overcomplicating the solution process.