Question:In the xy-plane, the graph of the equation 5x + 8y = 12 is translated 3 units to the right....

1. TRANSLATE the transformation information

• Given information:
  • Original equation: \(5\mathrm{x} + 8\mathrm{y} = 12\)
  • Translation: 3 units to the right
  • Need to find: y-coordinate of new y-intercept
• What "3 units to the right" means mathematically: replace every x with \(\mathrm{x} - 3\)

2. INFER the solution strategy

• To translate the graph, we need to transform the equation
• After getting the new equation, we'll find its y-intercept by setting \(\mathrm{x} = 0\)

3. SIMPLIFY the equation transformation

• Replace x with \(\mathrm{x} - 3\) in the original equation:
  \(5(\mathrm{x} - 3) + 8\mathrm{y} = 12\)
• Expand:
  \(5\mathrm{x} - 15 + 8\mathrm{y} = 12\)
• Rearrange:
  \(5\mathrm{x} + 8\mathrm{y} = 27\)

4. INFER how to find the y-intercept

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

5. SIMPLIFY to find the final answer

• Substitute \(\mathrm{x} = 0\) into \(5\mathrm{x} + 8\mathrm{y} = 27\):
  \(5(0) + 8\mathrm{y} = 27\)
  \(8\mathrm{y} = 27\)
  \(\mathrm{y} = \frac{27}{8}\)

Answer: \(\frac{27}{8}\) (or \(3.375\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the direction of translation and replace x with \(\mathrm{x} + 3\) instead of \(\mathrm{x} - 3\), thinking "right" means "positive."

This leads to the equation \(5(\mathrm{x} + 3) + 8\mathrm{y} = 12\), which simplifies to \(5\mathrm{x} + 8\mathrm{y} = -3\). Setting \(\mathrm{x} = 0\) gives \(\mathrm{y} = -\frac{3}{8}\), leading to confusion since this doesn't match typical answer choices.

Second Most Common Error:

Conceptual confusion about y-intercept: Students remember that intercepts involve setting a variable to zero, but mistakenly set \(\mathrm{y} = 0\) instead of \(\mathrm{x} = 0\).

When they set \(\mathrm{y} = 0\) in \(5\mathrm{x} + 8\mathrm{y} = 27\), they get \(\mathrm{x} = \frac{27}{5}\), which is the x-intercept rather than the y-intercept. This causes them to get stuck since the question asks for a y-coordinate.

The Bottom Line:

This problem requires careful attention to translation direction and clear understanding of what y-intercept means. The key insight is that horizontal translations affect the x-variable in a counterintuitive way: moving right means subtracting from x.