What is the y-coordinate of the y-intercept of the graph of x/3 - 4y/7 = -28/9 in the xy-plane?

1. TRANSLATE the problem information

• Given: Linear equation \(\frac{\mathrm{x}}{3} - \frac{4\mathrm{y}}{7} = -\frac{28}{9}\)
• Find: y-coordinate of y-intercept
• What this means: Find the y-value where the graph crosses the y-axis (where \(\mathrm{x} = 0\))

2. INFER the solution approach

• To find y-intercept, substitute \(\mathrm{x} = 0\) into the equation
• This will give us an equation with only y that we can solve

3. SIMPLIFY by substituting and solving

• Substitute \(\mathrm{x} = 0\): \(\frac{0}{3} - \frac{4\mathrm{y}}{7} = -\frac{28}{9}\)
• This simplifies to: \(-\frac{4\mathrm{y}}{7} = -\frac{28}{9}\)
• Multiply both sides by \(-\frac{7}{4}\): \(\mathrm{y} = \left(-\frac{28}{9}\right) \times \left(-\frac{7}{4}\right)\)

4. SIMPLIFY the fraction multiplication

• \(\mathrm{y} = \frac{28 \times 7}{9 \times 4} = \frac{196}{36}\)
• Reduce to lowest terms: Both 196 and 36 are divisible by 4
• \(\mathrm{y} = \frac{49}{9}\)

Answer: \(\frac{49}{9}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not understand that "y-intercept" means setting \(\mathrm{x} = 0\), instead trying to solve for when \(\mathrm{y} = 0\) (which would give the x-intercept).

This fundamental misunderstanding leads them to solve \(0 - \frac{4\mathrm{y}}{7} = -\frac{28}{9}\) incorrectly or attempt to find where the equation equals zero, causing confusion and potentially guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make arithmetic errors when multiplying fractions or fail to properly simplify \(\frac{196}{36}\).

Common mistakes include getting the wrong numerator/denominator when multiplying \(\left(-\frac{28}{9}\right) \times \left(-\frac{7}{4}\right)\), or stopping at \(\frac{196}{36}\) without reducing to \(\frac{49}{9}\). Since this is a student response question accepting equivalent forms, this might still be acceptable, but could indicate incomplete mathematical understanding.

The Bottom Line:

The key challenge is recognizing that y-intercept specifically requires setting \(\mathrm{x} = 0\), then carefully executing fraction operations without computational errors.