ax + by = 726x + 2by = 56In the given system of equations, a and b are constants. The...

1. TRANSLATE the problem information

• Given information:
  • Two equations: \(\mathrm{ax + by = 72}\) and \(\mathrm{6x + 2by = 56}\)
  • The graphs intersect at point \(\mathrm{(4, y)}\)
  • Need to find the value of a

• What this tells us: Since the graphs intersect at \(\mathrm{(4, y)}\), this point must satisfy both equations simultaneously.

2. INFER the solution approach

• Key insight: If \(\mathrm{(4, y)}\) lies on both graphs, then \(\mathrm{x = 4}\) must work in both equations
• Strategy: Substitute \(\mathrm{x = 4}\) into both equations to create a system we can solve for a

3. SIMPLIFY by substituting x = 4 into both equations

• First equation: \(\mathrm{a(4) + by = 72}\) → \(\mathrm{4a + by = 72}\)
• Second equation: \(\mathrm{6(4) + 2by = 56}\) → \(\mathrm{24 + 2by = 56}\)

4. SIMPLIFY the second equation to find by

• From \(\mathrm{24 + 2by = 56}\):
• \(\mathrm{2by = 56 - 24 = 32}\)
• \(\mathrm{by = 16}\)

5. SIMPLIFY to find a using the first equation

• Substitute \(\mathrm{by = 16}\) into \(\mathrm{4a + by = 72}\):
• \(\mathrm{4a + 16 = 72}\)
• \(\mathrm{4a = 56}\)
• \(\mathrm{a = 14}\)

Answer: D. 14

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students don't understand what "intersection point" means mathematically. They might think they need to find the y-coordinate first or try to solve the system without using the given intersection point.

This leads to attempting complex elimination or substitution methods on the original system, getting confused by the unknown y-value, and abandoning systematic solution for guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly understand the intersection concept but make calculation errors, particularly when manipulating the equation \(\mathrm{24 + 2by = 56}\) or when substituting back to find a.

Common mistakes include: \(\mathrm{2by = 24}\) (instead of 32), or \(\mathrm{by = 32}\) (instead of 16), leading to incorrect values like \(\mathrm{a = 3}\) or \(\mathrm{a = 6}\). This may lead them to select Choice A (3) or Choice C (6).

The Bottom Line:

The key breakthrough is recognizing that "intersection point" gives you direct information to substitute into both equations. Students who miss this translation step often overcomplicate the problem and lose their way in unnecessary algebraic manipulations.