If the graph of 27x + 33y = 297 is shifted down 5 units in the xy-plane, what is the...

1. TRANSLATE the transformation information

• Given: Graph of \(27x + 33y = 297\) is shifted down 5 units
• What this means: Replace \(y\) with \((y + 5)\) in the original equation

2. INFER the approach for finding y-intercepts

• To find any y-intercept: set \(x = 0\) in the equation
• But first: we need the equation of the transformed graph

3. SIMPLIFY to get the new equation

• Start with: \(27x + 33(y + 5) = 297\)
• Distribute: \(27x + 33y + 165 = 297\)
• Subtract 165: \(27x + 33y = 132\)

4. APPLY the y-intercept method

• Set \(x = 0\): \(27(0) + 33y = 132\)
• Solve: \(33y = 132\)
• Therefore: \(y = 4\)

Answer: A. \((0, 4)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often confuse the direction of vertical shifts. They might think "shifted down 5 units" means subtracting 5 from y, leading them to write \(27x + 33(y - 5) = 297\) instead of \(27x + 33(y + 5) = 297\).

Following this incorrect transformation:

• \(27x + 33(y - 5) = 297\)
• \(27x + 33y - 165 = 297\)
• \(27x + 33y = 462\)
• Setting \(x = 0\): \(y = 14\)

This may lead them to select Choice C. \((0, 14)\).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students might correctly set up \(27x + 33(y + 5) = 297\) but make algebraic errors when expanding or solving.

A typical mistake is incorrectly expanding to get \(27x + 33y + 33 = 297\) (using \(33 \times 1\) instead of \(33 \times 5\)), which leads to \(27x + 33y = 264\) and ultimately \(y = 8\). While this specific value isn't among the choices, such calculation errors cause students to get confused and guess.

The Bottom Line:

This problem tests whether students can correctly translate vertical transformations into algebraic form. The key insight is that "down 5 units" means every y-value becomes 5 units smaller, so to compensate in the equation, we need y to become 5 units larger—hence \(y + 5\).