The variables x and y satisfy the equation 2y - x = 8. Which table gives three values of x...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{2y - x = 8}\)
• Need to find: Which table correctly shows x and y pairs that satisfy this equation

2. INFER the approach

• Since we have one equation with two variables, we should solve for one variable in terms of the other
• Solving for y in terms of x will let us calculate the y-value for any given x-value
• We can then check which table matches our calculated values

3. SIMPLIFY the equation to solve for y

• Start with: \(\mathrm{2y - x = 8}\)
• Add x to both sides: \(\mathrm{2y = 8 + x}\)
• Divide both sides by 2: \(\mathrm{y = \frac{8 + x}{2}}\)

4. SIMPLIFY by substituting the x-values from the tables

For \(\mathrm{x = 0}\):
\(\mathrm{y = \frac{8 + 0}{2}}\)
\(\mathrm{y = \frac{8}{2}}\)
\(\mathrm{y = 4}\)

For \(\mathrm{x = 2}\):
\(\mathrm{y = \frac{8 + 2}{2}}\)
\(\mathrm{y = \frac{10}{2}}\)
\(\mathrm{y = 5}\)

For \(\mathrm{x = 4}\):
\(\mathrm{y = \frac{8 + 4}{2}}\)
\(\mathrm{y = \frac{12}{2}}\)
\(\mathrm{y = 6}\)

5. INFER which table matches our results

• Our calculated pairs: \(\mathrm{(0, 4), (2, 5), (4, 6)}\)
• Only choice (A) contains exactly these pairs

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making sign errors when rearranging the equation

Students might incorrectly rearrange \(\mathrm{2y - x = 8}\) as \(\mathrm{2y = 8 - x}\) instead of \(\mathrm{2y = 8 + x}\). This gives them \(\mathrm{y = \frac{8 - x}{2}}\), leading to:

• When \(\mathrm{x = 0}\): \(\mathrm{y = 4}\) ✓ (happens to be correct)
• When \(\mathrm{x = 2}\): \(\mathrm{y = 3}\) ✗
• When \(\mathrm{x = 4}\): \(\mathrm{y = 2}\) ✗

This may lead them to select Choice B.

Second Most Common Error:

Incomplete SIMPLIFY execution: Forgetting to divide by the coefficient

Students correctly get \(\mathrm{2y = 8 + x}\) but then forget to divide the entire right side by 2, thinking \(\mathrm{y = 8 + x}\). This gives them:

• When \(\mathrm{x = 0}\): \(\mathrm{y = 8}\) ✗
• When \(\mathrm{x = 2}\): \(\mathrm{y = 10}\) ✗
• When \(\mathrm{x = 4}\): \(\mathrm{y = 12}\) ✗

None of these match exactly, but students might be drawn to Choice C which shows \(\mathrm{y = 8 - x}\) instead.

The Bottom Line:

This problem tests careful algebraic manipulation. The key insight is that every step in solving for y must be applied to the entire expression, and sign changes must be tracked precisely. Students who rush through the algebra often make systematic errors that produce consistent but wrong patterns in their calculations.