Question:4x = 20x + 3y = 17The solution to the given system of equations is \(\mathrm{(x, y)}\). What is the...

1. TRANSLATE the problem information

• Given information:
  • First equation: \(\mathrm{4x = 20}\)
  • Second equation: \(\mathrm{x + 3y = 17}\)
  • Need to find: the value of y

2. INFER the solution strategy

• Notice the first equation only contains x and can be solved directly
• Strategy: Solve the first equation for x, then substitute that value into the second equation
• This substitution method works well when one equation is simple to solve

3. SIMPLIFY the first equation to find x

• Start with: \(\mathrm{4x = 20}\)
• Divide both sides by 4: \(\mathrm{x = \frac{20}{4} = 5}\)

4. SIMPLIFY by substituting \(\mathrm{x = 5}\) into the second equation

• Replace x with 5 in: \(\mathrm{x + 3y = 17}\)
• Get: \(\mathrm{5 + 3y = 17}\)
• Subtract 5 from both sides: \(\mathrm{3y = 17 - 5 = 12}\)
• Divide both sides by 3: \(\mathrm{y = \frac{12}{3} = 4}\)

Answer: 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make basic arithmetic mistakes when solving simple equations

• Incorrectly calculating \(\mathrm{20 ÷ 4 = 6}\) instead of 5
• Or making errors like \(\mathrm{17 - 5 = 11}\) instead of 12
• Or calculating \(\mathrm{12 ÷ 3 = 5}\) instead of 4

These arithmetic slips lead them to select Choice A (3) or Choice C (5) instead of the correct answer.

Second Most Common Error:

Poor INFER reasoning: Students attempt to solve both equations simultaneously or use elimination method unnecessarily

• They might try to manipulate both equations instead of recognizing the simple substitution opportunity
• This leads to more complex algebra and higher chance of errors
• May cause confusion about which variable to solve for first

This complicates the solution process and often leads to calculation errors, causing them to select Choice D (12) or abandon systematic solving and guess.

The Bottom Line:

This problem rewards students who can spot the easiest path (solve the simpler equation first) and execute basic arithmetic accurately. The key insight is recognizing when substitution is the natural choice.