The given equation relates the positive numbers b, x, and y. Which equation correctly expresses x in terms of b...

1. INFER the solution strategy

• Goal: Isolate x on one side of the equation
• Strategy: Eliminate fractions by multiplying, then isolate x by dividing
• Start with: \(\frac{1}{7\mathrm{b}} = \frac{11\mathrm{x}}{\mathrm{y}}\)

2. SIMPLIFY by eliminating y from the denominator

• Multiply both sides by y:
  • Left side: \(\mathrm{y} \cdot \frac{1}{7\mathrm{b}} = \frac{\mathrm{y}}{7\mathrm{b}}\)
  • Right side: \(\mathrm{y} \cdot \frac{11\mathrm{x}}{\mathrm{y}} = 11\mathrm{x}\)
• Result: \(\frac{\mathrm{y}}{7\mathrm{b}} = 11\mathrm{x}\)

3. SIMPLIFY by isolating x

• Divide both sides by 11:
  • Left side: \(\frac{\mathrm{y}/(7\mathrm{b})}{11} = \frac{\mathrm{y}}{7\mathrm{b} \cdot 11} = \frac{\mathrm{y}}{77\mathrm{b}}\)
  • Right side: \(\frac{11\mathrm{x}}{11} = \mathrm{x}\)
• Result: \(\frac{\mathrm{y}}{77\mathrm{b}} = \mathrm{x}\)

Answer: C. \(\mathrm{x} = \frac{\mathrm{y}}{77\mathrm{b}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make errors when working with compound fractions or combining denominators.

Some students might incorrectly handle the fraction operations, perhaps multiplying denominators incorrectly (getting \(7\mathrm{b} \times 11 = 77\mathrm{b}\) right) but placing variables in wrong positions, or confusing the order of operations when dealing with \(\frac{\mathrm{y}}{7\mathrm{b}} \div 11\). This computational confusion can lead them to arrangements like \(\frac{7\mathrm{by}}{11}\).

This may lead them to select Choice A (\(\mathrm{x} = \frac{7\mathrm{by}}{11}\)).

Second Most Common Error:

Poor INFER reasoning about equation manipulation: Students attempt cross-multiplication without properly understanding the structure of the equation.

Instead of systematically clearing denominators, they might try to "cross multiply" the original equation \(\frac{1}{7\mathrm{b}} = \frac{11\mathrm{x}}{\mathrm{y}}\), incorrectly treating it like a proportion and getting confused about which terms go where. This can result in expressions that mix up the positions of the variables and coefficients.

This may lead them to select Choice D (\(\mathrm{x} = 77\mathrm{by}\)).

The Bottom Line:

This problem requires careful attention to fraction operations and systematic algebraic manipulation. Students who rush through the steps or lack confidence with compound fractions are most likely to make computational errors that lead to incorrect arrangements of the variables and coefficients.