14j + 5k = mThe given equation relates the numbers j, k, and m. Which equation correctly expresses k in...

1. INFER the solution strategy

• Given: \(14\mathrm{j} + 5\mathrm{k} = \mathrm{m}\)
• Goal: Express k in terms of j and m
• Strategy: Isolate k by "undoing" the operations in reverse order
• Since k is being multiplied by 5, then added to 14j, we need to subtract 14j first, then divide by 5

2. SIMPLIFY by subtracting 14j from both sides

• \(14\mathrm{j} + 5\mathrm{k} = \mathrm{m}\)
• \(14\mathrm{j} + 5\mathrm{k} - 14\mathrm{j} = \mathrm{m} - 14\mathrm{j}\)
• \(5\mathrm{k} = \mathrm{m} - 14\mathrm{j}\)

3. SIMPLIFY by dividing both sides by 5

• \(5\mathrm{k} = \mathrm{m} - 14\mathrm{j}\)
• \(\mathrm{k} = \frac{\mathrm{m} - 14\mathrm{j}}{5}\)

Answer: A. k = (m - 14j)/5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when rearranging terms, writing \((14\mathrm{j} - \mathrm{m})\) instead of \((\mathrm{m} - 14\mathrm{j})\).

When subtracting 14j from both sides, they might think: "I need to get 14j on the right side, so it becomes 14j - m." They forget that subtracting 14j from m gives m - 14j, not 14j - m.

This leads them to select Choice C \(\left(\frac{14\mathrm{j} - \mathrm{m}}{5}\right)\)

Second Most Common Error:

Poor INFER reasoning about order of operations: Students correctly subtract 14j but forget that the entire expression \((\mathrm{m} - 14\mathrm{j})\) must be divided by 5, not just the m term.

They might write \(\mathrm{k} = \frac{\mathrm{m}}{5} - 14\mathrm{j}\), thinking they can distribute the division. They don't recognize that when \(5\mathrm{k} = (\mathrm{m} - 14\mathrm{j})\), dividing by 5 means \(\mathrm{k} = \frac{\mathrm{m} - 14\mathrm{j}}{5}\).

This leads them to select Choice B \(\left(\frac{1}{5}\mathrm{m} - 14\mathrm{j}\right)\)

The Bottom Line:

Success requires careful attention to signs and understanding that division by 5 applies to the entire expression \((\mathrm{m} - 14\mathrm{j})\), not individual terms.