The equation x = 7y - z relates the variables x, y, and z. Which of the following correctly expresses...

1. TRANSLATE the problem requirements

• Given information:
  • Equation: \(\mathrm{x = 7y - z}\)
  • Need to find: \(\mathrm{x + z}\) expressed in terms of y

• What this means: We need to get \(\mathrm{x + z}\) by itself on one side of an equation, with only y terms on the other side.

2. INFER the solution strategy

• Key insight: Since we have \(\mathrm{x = 7y - z}\), and we want \(\mathrm{x + z}\), we should add z to both sides
• This will put \(\mathrm{x + z}\) on the left side and eliminate the negative z on the right side

3. SIMPLIFY through algebraic manipulation

• Start with: \(\mathrm{x = 7y - z}\)
• Add z to both sides: \(\mathrm{x + z = 7y - z + z}\)
• Combine like terms on the right: \(\mathrm{x + z = 7y + 0}\)
• Final result: \(\mathrm{x + z = 7y}\)

Answer: A. 7y

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the strategy of adding z to both sides to create the desired expression \(\mathrm{x + z}\).

Instead, they might try to solve for individual variables first (finding y in terms of x and z, or z in terms of x and y), which makes the problem much more complicated than necessary. This leads to confusion and often causes them to guess rather than work systematically.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the strategy but make sign errors during algebraic manipulation.

They might incorrectly think that \(\mathrm{7y - z + z}\) equals \(\mathrm{7y - 2z}\) or get confused about whether the z terms truly cancel out. This computational error may lead them to select Choice D (\(\mathrm{14y}\)) if they somehow double the result, or get stuck and guess.

The Bottom Line:

This problem tests whether students can manipulate equations strategically rather than just solve for individual variables. The key insight is recognizing that you can directly create the expression you need (\(\mathrm{x + z}\)) through a simple algebraic operation, rather than solving the system step by step.