The mean (average) of two real numbers x and y is t. Which equation correctly expresses x in terms of...

1. TRANSLATE the problem information

• Given information:
  • The mean (average) of two real numbers x and y is t
• What this tells us: We can write this as the equation \(\frac{\mathrm{x + y}}{2} = \mathrm{t}\)

2. SIMPLIFY to isolate x

• Start with: \(\frac{\mathrm{x + y}}{2} = \mathrm{t}\)
• Clear the denominator by multiplying both sides by 2: \(\mathrm{x + y = 2t}\)
• Subtract y from both sides to isolate x: \(\mathrm{x = 2t - y}\)

Answer: D. x = 2t - y

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students incorrectly clear the denominator or make sign errors during algebraic manipulation.

Many students struggle with the step of multiplying both sides by 2. They might only multiply one side, or forget to distribute the multiplication properly. For example, they might incorrectly get \(\mathrm{x + y = t}\) (forgetting to multiply the right side by 2), leading to \(\mathrm{x = t - y}\). This may lead them to select Choice B \(\mathrm{(x = t - \frac{y}{2})}\) after further algebraic confusion.

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students misunderstand what "mean" means algebraically or set up the initial equation incorrectly.

Some students know that mean involves division by 2, but they might write something like \(\mathrm{x + y = \frac{t}{2}}\) instead of \(\frac{\mathrm{x + y}}{2} = \mathrm{t}\). This fundamental setup error cascades through their algebraic work and leads to answers like Choice A \(\mathrm{(x = \frac{t}{2} - y)}\).

The Bottom Line:

This problem tests whether students can accurately translate a verbal description of mean into a mathematical equation and then perform reliable algebraic manipulation. The key insight is recognizing that clearing fractions early makes the algebra much cleaner.