y = -1/5xy = 1/2xThe solution to the given system of equations is \(\mathrm{(x, y)}\). What is the value of...

1. TRANSLATE the problem information

• Given system of equations:
  • \(\mathrm{y = -\frac{1}{5}x}\)
  • \(\mathrm{y = \frac{1}{2}x}\)
• We need to find the value of x where both equations are satisfied

2. TRANSLATE the solution strategy

• Since both equations equal y, we can set the right-hand sides equal to each other
• This gives us: \(\mathrm{-\frac{1}{5}x = \frac{1}{2}x}\)

3. SIMPLIFY by solving for x

• Move all x terms to one side: \(\mathrm{-\frac{1}{5}x - \frac{1}{2}x = 0}\)
• Find common denominator of 10:
  • \(\mathrm{-\frac{1}{5}x = -\frac{2}{10}x}\)
  • \(\mathrm{\frac{1}{2}x = \frac{5}{10}x}\)
• Combine: \(\mathrm{-\frac{2}{10}x - \frac{5}{10}x = -\frac{7}{10}x = 0}\)
• Since \(\mathrm{-\frac{7}{10}x = 0}\), we get \(\mathrm{x = 0}\)

4. Verify the solution

• If \(\mathrm{x = 0}\): \(\mathrm{y = -\frac{1}{5}(0) = 0}\) and \(\mathrm{y = \frac{1}{2}(0) = 0}\)
• Both equations give the same y-value, confirming our solution

Answer: B. 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make fraction arithmetic errors when combining \(\mathrm{-\frac{1}{5}x}\) and \(\mathrm{\frac{1}{2}x}\). They might incorrectly find the common denominator or make sign errors during the subtraction process. For example, they might get \(\mathrm{-\frac{1}{5}x - \frac{1}{2}x = -\frac{3}{7}x}\) instead of \(\mathrm{-\frac{7}{10}x}\), leading to an incorrect final answer.

This may lead them to select Choice A (-5), Choice C (2), or Choice D (7) depending on their specific calculation errors.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might not recognize that they can set the two expressions equal to each other. Instead, they might try to solve each equation individually for specific values, not understanding that they need to find the intersection point of the two lines.

This leads to confusion and guessing among the available choices.

The Bottom Line:

This problem tests whether students understand the fundamental principle that solutions to systems of equations represent points where all equations are simultaneously satisfied, combined with careful fraction arithmetic execution.