The graph of 9x - 10y = 19 is translated down 4 units in the xy-plane. What is the x-coordinate...

1. TRANSLATE the transformation information

• Given information:
  • Original equation: \(9x - 10y = 19\)
  • Graph is translated down 4 units
  • Need to find x-coordinate of x-intercept

• What "translated down 4 units" means mathematically: Replace \(y\) with \((y + 4)\) in the equation

2. APPLY the translation to the equation

• Original: \(9x - 10y = 19\)
• After translation: \(9x - 10(y + 4) = 19\)

3. SIMPLIFY the new equation

• Expand: \(9x - 10y - 40 = 19\)
• Add 40 to both sides: \(9x - 10y = 59\)

4. INFER how to find the x-intercept

• The x-intercept occurs where the graph crosses the x-axis
• At this point, \(y = 0\)

5. SUBSTITUTE and solve

• Set \(y = 0\): \(9x - 10(0) = 59\)
• Simplify: \(9x = 59\)
• Divide by 9: \(x = \frac{59}{9}\)

Answer: \(\frac{59}{9}\) (or 6.555 or 6.556)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly interpret "translated down 4 units" as subtracting 4 from the constant term, getting \(9x - 10y = 15\) instead of properly replacing \(y\) with \((y + 4)\).

When they find the x-intercept of \(9x - 10y = 15\), they get \(x = \frac{15}{9} = \frac{5}{3} ≈ 1.67\). This leads to confusion since this value doesn't match any typical answer format, causing them to guess.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(9x - 10(y + 4) = 19\) but make algebraic errors when expanding, such as getting \(9x - 10y + 40 = 19\) (wrong sign on the 40).

This leads them to \(9x - 10y = -21\), and setting \(y = 0\) gives \(x = \frac{-21}{9} = \frac{-7}{3}\). Since this is negative and doesn't make sense in their mental model, this causes them to abandon systematic solution and guess.

The Bottom Line:

This problem tests whether students can correctly translate geometric transformations into algebraic operations and then follow through with accurate algebraic manipulation.