The graph of 5x + 3y = 12 is translated to the right by 2 units in the xy-plane. What...

1. TRANSLATE the transformation description

• Given information:
  • Original equation: \(5\mathrm{x} + 3\mathrm{y} = 12\)
  • Translation: "right by 2 units"

• TRANSLATE tells us: Moving right by 2 units means replacing \(\mathrm{x}\) with \(\mathrm{x} - 2\) in our equation

2. INFER the solution strategy

• We need to transform the equation first, then find where it crosses the y-axis
• Strategy: Apply the transformation, then set \(\mathrm{x} = 0\)

3. SIMPLIFY the transformed equation

• Start with: \(5\mathrm{x} + 3\mathrm{y} = 12\)
• Replace \(\mathrm{x}\) with \(\mathrm{x} - 2\): \(5(\mathrm{x} - 2) + 3\mathrm{y} = 12\)
• Expand: \(5\mathrm{x} - 10 + 3\mathrm{y} = 12\)
• Add 10 to both sides: \(5\mathrm{x} + 3\mathrm{y} = 22\)

4. INFER how to find the y-intercept

• The y-intercept occurs where the graph crosses the y-axis
• This happens when \(\mathrm{x} = 0\)

5. SIMPLIFY to find the y-coordinate

• Substitute \(\mathrm{x} = 0\) into \(5\mathrm{x} + 3\mathrm{y} = 22\)
• \(5(0) + 3\mathrm{y} = 22\)
• \(3\mathrm{y} = 22\)
• \(\mathrm{y} = \frac{22}{3}\)

Answer: B) 22/3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may translate "right by 2 units" incorrectly as replacing \(\mathrm{x}\) with \(\mathrm{x} + 2\) instead of \(\mathrm{x} - 2\). This counterintuitive relationship confuses many students.

Following this incorrect translation:

• \(5(\mathrm{x} + 2) + 3\mathrm{y} = 12\)
• \(5\mathrm{x} + 10 + 3\mathrm{y} = 12\)
• \(5\mathrm{x} + 3\mathrm{y} = 2\)

Setting \(\mathrm{x} = 0\): \(3\mathrm{y} = 2\), so \(\mathrm{y} = \frac{2}{3}\)

This doesn't match any answer choice, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the transformation but make algebraic errors when expanding or rearranging.

For example, they might write:

• \(5(\mathrm{x} - 2) + 3\mathrm{y} = 12\)
• \(5\mathrm{x} - 10 + 3\mathrm{y} = 12\)
• \(5\mathrm{x} + 3\mathrm{y} = 12 - 10 = 2\) (incorrect sign)

This leads to \(\mathrm{y} = \frac{2}{3}\), which again doesn't match the choices.

The Bottom Line:

The trickiest part is remembering that moving right by h units requires replacing \(\mathrm{x}\) with \(\mathrm{x} - \mathrm{h}\), not \(\mathrm{x} + \mathrm{h}\). This counterintuitive relationship is the key conceptual hurdle that causes most student errors.