4x + y = 9z The given equation relates the variables x, y, and z. Which equation correctly expresses x...

1. TRANSLATE the problem requirement

• Given equation: \(\mathrm{4x + y = 9z}\)
• Goal: Express \(\mathrm{x}\) in terms of \(\mathrm{y}\) and \(\mathrm{z}\) (isolate \(\mathrm{x}\) on one side)

2. SIMPLIFY by eliminating the y term from the x side

• Subtract \(\mathrm{y}\) from both sides:
  \(\mathrm{4x + y - y = 9z - y}\)
  \(\mathrm{4x = 9z - y}\)

3. SIMPLIFY by eliminating the coefficient of x

• Divide both sides by 4:
  \(\mathrm{x = \frac{9z - y}{4}}\)

4. TRANSLATE to match with answer choices

• Look for the choice that matches \(\mathrm{x = \frac{9z - y}{4}}\)
• Pay careful attention to parentheses and order of operations

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misread answer choice (A) as matching their work because they don't carefully consider order of operations. They see "\(\mathrm{9z - \frac{y}{4}}\)" and think it's the same as "\(\mathrm{\frac{9z - y}{4}}\)", when actually "\(\mathrm{9z - \frac{y}{4}}\)" means "\(\mathrm{9z - \left(\frac{y}{4}\right)}\)".

This leads them to select Choice A (\(\mathrm{x = 9z - \frac{y}{4}}\)) even when their algebraic work was correct.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make an algebraic error in the isolation process, such as adding \(\mathrm{y}\) to both sides instead of subtracting it, or forgetting to divide the entire right side by 4.

For example, if they get \(\mathrm{4x = 9z - y}\) but then write \(\mathrm{x = \frac{9z}{4} - y}\) (dividing only part of the right side), this leads them to select Choice B (\(\mathrm{x = \frac{9z}{4} - y}\)).

The Bottom Line:

This problem tests both algebraic manipulation skills and careful reading of mathematical expressions. Success requires not just knowing how to isolate variables, but also understanding how parentheses and order of operations affect the meaning of algebraic expressions.