The equation 40x + 20y = 160 represents the number of sweaters, x, and number of shirts, y, that Yesenia...

1. TRANSLATE the problem information

• Given equation: \(40\mathrm{x} + 20\mathrm{y} = 160\) represents the cost relationship
• Given information: Yesenia purchased 2 sweaters, so \(\mathrm{x} = 2\)
• What we need to find: number of shirts \(\mathrm{y}\)

2. SIMPLIFY by substituting the known value

• Substitute \(\mathrm{x} = 2\) into the equation:
  \(40(2) + 20\mathrm{y} = 160\)
• Calculate: \(40 \times 2 = 80\)
  So: \(80 + 20\mathrm{y} = 160\)

3. SIMPLIFY to isolate the variable y

• Subtract 80 from both sides:
  \(80 + 20\mathrm{y} - 80 = 160 - 80\)
  \(20\mathrm{y} = 80\)
• Divide both sides by 20:
  \(\mathrm{y} = 80 \div 20 = 4\)

Answer: B. 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making arithmetic errors during the multi-step calculation process.

Students might incorrectly calculate \(40 \times 2 = 60\) instead of 80, leading to \(60 + 20\mathrm{y} = 160\), then \(20\mathrm{y} = 100\), so \(\mathrm{y} = 5\). Since 5 isn't an answer choice, this leads to confusion and guessing.

Alternatively, they might make errors when subtracting: \(160 - 80 = 70\) instead of 80, getting \(\mathrm{y} = 3.5\), and then rounding to select Choice A (3).

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding which variable to substitute or confusing the setup.

Some students might try to substitute \(\mathrm{y} = 2\) instead of \(\mathrm{x} = 2\), not recognizing that "2 sweaters" means \(\mathrm{x} = 2\). This leads to \(40\mathrm{x} + 20(2) = 160\), giving \(40\mathrm{x} + 40 = 160\), so \(40\mathrm{x} = 120\), and \(\mathrm{x} = 3\). They might then incorrectly select Choice A (3) thinking this represents the number of shirts.

The Bottom Line:

This problem requires careful attention to which variable represents which quantity, followed by systematic algebraic manipulation without arithmetic mistakes.