-{x - wy = -337}2x - wy = 47In the given system of equations, w is a constant. In the...

1. TRANSLATE the intersection information

• Given information:
  • System: \(-\mathrm{x} - \mathrm{wy} = -337\) and \(2\mathrm{x} - \mathrm{wy} = 47\)
  • Graphs intersect at point \((\mathrm{q}, 19)\)
• What this tells us: The point \((\mathrm{q}, 19)\) must satisfy both equations since it lies on both lines

2. INFER the solution strategy

• We have two equations with three unknowns (x, y, w), but we know the intersection point
• Strategy: Substitute the known point \((\mathrm{q}, 19)\) into both equations to create a system with only q and w
• This will allow us to solve for w

3. TRANSLATE the first equation with the intersection point

• Substitute x = q and y = 19 into \(-\mathrm{x} - \mathrm{wy} = -337\):
  \(-\mathrm{q} - \mathrm{w}(19) = -337\)
  \(-\mathrm{q} - 19\mathrm{w} = -337\)

4. SIMPLIFY to express q in terms of w

• From \(-\mathrm{q} - 19\mathrm{w} = -337\):
  \(-\mathrm{q} = -337 + 19\mathrm{w}\)
  \(\mathrm{q} = 337 - 19\mathrm{w}\)

5. TRANSLATE the second equation with the intersection point

• Substitute x = q and y = 19 into \(2\mathrm{x} - \mathrm{wy} = 47\):
  \(2\mathrm{q} - \mathrm{w}(19) = 47\)
  \(2\mathrm{q} - 19\mathrm{w} = 47\)

6. INFER the substitution step

• We now have \(\mathrm{q} = 337 - 19\mathrm{w}\) from step 4
• Substitute this expression for q into the equation from step 5

7. SIMPLIFY to solve for w

• Substitute \(\mathrm{q} = 337 - 19\mathrm{w}\) into \(2\mathrm{q} - 19\mathrm{w} = 47\):
  \(2(337 - 19\mathrm{w}) - 19\mathrm{w} = 47\)
• Distribute:
  \(674 - 38\mathrm{w} - 19\mathrm{w} = 47\)
• Combine like terms:
  \(674 - 57\mathrm{w} = 47\)
• Solve:
  \(-57\mathrm{w} = 47 - 674 = -627\)
• Therefore:
  \(\mathrm{w} = -627/(-57) = 11\)

Answer: 11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not recognizing that "intersection point \((\mathrm{q}, 19)\)" means this point must satisfy both equations simultaneously.

Students might try to solve the system using elimination or other methods without using the intersection point information, leading to confusion since they have more unknowns than they can handle. This causes them to get stuck and abandon systematic solution, often guessing randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors when combining like terms or solving the linear equation.

For example, incorrectly combining \(-38\mathrm{w} - 19\mathrm{w}\) as \(-47\mathrm{w}\) instead of \(-57\mathrm{w}\), or making sign errors when isolating w. This leads to an incorrect value of w that doesn't match any reasonable answer, causing confusion and potentially random answer selection.

The Bottom Line:

This problem tests whether students can connect geometric information (intersection point) with algebraic requirements (point satisfies equations). The key insight is that knowing where two lines intersect gives you powerful constraints that can determine unknown parameters in the system.