The equation 46 = 2a + 2b gives the relationship between the side lengths a and b of a certain...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(46 = 2\mathrm{a} + 2\mathrm{b}\) (relationship between parallelogram sides)
  • Value: \(\mathrm{a} = 9\)
  • Find: value of b

2. INFER the solution strategy

• Since we know the value of a and have an equation with both a and b, we should substitute the known value to solve for the unknown
• Strategy: Substitute \(\mathrm{a} = 9\) into the equation, then solve for b

3. SIMPLIFY through substitution and algebraic steps

• Substitute \(\mathrm{a} = 9\) into \(46 = 2\mathrm{a} + 2\mathrm{b}\):
  \(46 = 2(9) + 2\mathrm{b}\)
  \(46 = 18 + 2\mathrm{b}\)

• Subtract 18 from both sides:
  \(46 - 18 = 2\mathrm{b}\)
  \(28 = 2\mathrm{b}\)

• Divide both sides by 2:
  \(\mathrm{b} = 28/2 = 14\)

Answer: 14

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the calculation steps, particularly when subtracting 18 from 46 or dividing 28 by 2. They might calculate \(46 - 18 = 26\) instead of 28, leading to \(\mathrm{b} = 13\), or make errors in the division step.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what the equation represents or what they're solving for. They might try to solve for both a and b simultaneously, or get confused about which variable to substitute, leading to abandoning systematic solution and guessing.

The Bottom Line:

This problem tests fundamental algebraic substitution skills. Success requires careful arithmetic and systematic equation solving, with the main challenge being maintaining accuracy through multiple calculation steps.