7/2x + 6y = 25 5/2x + 6y = 23 The solution to the given system of equations is \((\mathrm{x},...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\frac{7}{2}\mathrm{x} + 6\mathrm{y} = 25\)
  • Second equation: \(\frac{5}{2}\mathrm{x} + 6\mathrm{y} = 23\)
  • Need to find: \(\frac{17}{2}\mathrm{x} + 18\mathrm{y}\)

• What this tells us: We need to find the value of a specific expression, not necessarily the individual values of x and y.

2. INFER the most efficient approach

• Key insight: Notice that \(\frac{17}{2}\mathrm{x} + 18\mathrm{y}\) looks like it could be formed by combining the left sides of our two equations
• Strategic decision: Instead of solving for x and y individually, let's see if we can manipulate the equations to directly get \(\frac{17}{2}\mathrm{x} + 18\mathrm{y}\)

3. SIMPLIFY by manipulating the equations strategically

• Multiply the second equation by 2:
  \(2\left(\frac{5}{2}\mathrm{x} + 6\mathrm{y}\right) = 2(23)\)
  \(\frac{10}{2}\mathrm{x} + 12\mathrm{y} = 46\)

• Now add this new equation to the first equation:
  \(\left(\frac{7}{2}\mathrm{x} + 6\mathrm{y}\right) + \left(\frac{10}{2}\mathrm{x} + 12\mathrm{y}\right) = 25 + 46\)

• Combine like terms:
  \(\left(\frac{7}{2}\mathrm{x} + \frac{10}{2}\mathrm{x}\right) + (6\mathrm{y} + 12\mathrm{y}) = 71\)
  \(\frac{17}{2}\mathrm{x} + 18\mathrm{y} = 71\)

Answer: D. 71

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students immediately try to solve for x and y individually using elimination or substitution, then substitute these values into \(\frac{17}{2}\mathrm{x} + 18\mathrm{y}\).

While this approach works, it's much longer and creates more opportunities for calculation errors. Students get \(\mathrm{x} = 2\) and \(\mathrm{y} = 3\), then calculate \(\frac{17}{2}(2) + 18(3) = 17 + 54 = 71\). Though they reach the correct answer, they miss the elegant direct approach and waste valuable time.

Second Most Common Error:

Poor SIMPLIFY execution: Students attempt the direct manipulation approach but make errors when multiplying the second equation by 2 or when adding the equations together.

For example, they might incorrectly get \(5\mathrm{x} + 12\mathrm{y} = 46\) instead of \(\frac{10}{2}\mathrm{x} + 12\mathrm{y} = 46\), or make arithmetic errors when adding \(25 + 46\). These computational mistakes lead to wrong final values, causing them to select Choice A (2) or Choice B (3) - which are actually the individual values of x and y.

The Bottom Line:

This problem rewards strategic thinking over computational grinding. The key insight is recognizing that you can directly construct the target expression \(\frac{17}{2}\mathrm{x} + 18\mathrm{y}\) through equation manipulation, avoiding the need to find individual variable values.