A formula relates the quantities j, k, and m: \(\mathrm{3m = 7k + 2(5j - k)}\). Which equation correctly expresses...

1. TRANSLATE the problem information

• Given: \(\mathrm{3m = 7k + 2(5j - k)}\)
• Find: k expressed in terms of j and m (k = something with j and m)

2. INFER the approach

• We need to isolate k on one side of the equation
• First step: simplify the right side by distributing and combining like terms
• Then use inverse operations to get k by itself

3. SIMPLIFY by distributing the 2

• \(\mathrm{3m = 7k + 2(5j - k)}\)
• \(\mathrm{3m = 7k + 2(5j) + 2(-k)}\)
• \(\mathrm{3m = 7k + 10j - 2k}\)

4. SIMPLIFY by combining like terms

• The k terms are 7k and -2k
• \(\mathrm{7k - 2k = 5k}\)
• So: \(\mathrm{3m = 5k + 10j}\)

5. SIMPLIFY to isolate the k term

• Subtract 10j from both sides:
• \(\mathrm{3m - 10j = 5k}\)

6. SIMPLIFY to solve for k

• Divide both sides by 5:
• \(\mathrm{k = \frac{3m - 10j}{5}}\)

Answer: (A) k = (3m - 10j)/5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when distributing the 2, especially with the negative term 2(-k).

They might write: \(\mathrm{3m = 7k + 10j + 2k}\) instead of \(\mathrm{3m = 7k + 10j - 2k}\)

This gives them \(\mathrm{3m = 9k + 10j}\), leading to \(\mathrm{k = \frac{3m - 10j}{9}}\), which doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY reasoning: Students correctly distribute but make errors combining like terms, getting \(\mathrm{3m = 9k + 10j}\) instead of \(\mathrm{3m = 5k + 10j}\).

This leads them to \(\mathrm{k = \frac{3m - 10j}{9}}\), again not matching any choice, causing them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests careful algebraic manipulation through multiple steps. Students must maintain accuracy with signs and coefficients throughout the entire process to reach the correct form that matches the answer choices.