What is the y-coordinate of the y-intercept of the graph of 9x/7 = -5y/9 + 21 in the xy-plane?

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\frac{9\mathrm{x}}{7} = -\frac{5\mathrm{y}}{9} + 21\)
  • Need: y-coordinate of the y-intercept

• What this tells us: The y-intercept occurs where the graph crosses the y-axis, which means \(\mathrm{x} = 0\).

2. SIMPLIFY by substituting x = 0

• Substitute \(\mathrm{x} = 0\) into the equation:
  \(\frac{9(0)}{7} = -\frac{5\mathrm{y}}{9} + 21\)

• This gives us:
  \(0 = -\frac{5\mathrm{y}}{9} + 21\)

3. SIMPLIFY to solve for y

• Subtract 21 from both sides:
  \(0 - 21 = -\frac{5\mathrm{y}}{9} + 21 - 21\)
  \(-21 = -\frac{5\mathrm{y}}{9}\)

• Multiply both sides by -9:
  \((-21)(-9) = \left(-\frac{5\mathrm{y}}{9}\right)(-9)\)
  \(189 = 5\mathrm{y}\)

• Divide both sides by 5:
  \(\frac{189}{5} = \mathrm{y}\)

Answer: 189/5 or 37.8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not understand that "y-intercept" means the point where \(\mathrm{x} = 0\), or they might confuse it with x-intercept (where \(\mathrm{y} = 0\)).

Instead of substituting \(\mathrm{x} = 0\), they might substitute \(\mathrm{y} = 0\), leading to a completely different calculation that finds the x-intercept instead. This leads to confusion and potentially guessing on the final answer.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when working with the negative numbers and fractions, particularly when multiplying -21 by -9.

Common mistakes include:

• Getting the wrong sign: \((-21)(-9) = -189\) instead of +189
• Forgetting to multiply by the reciprocal correctly
• Making basic arithmetic errors with \(189 \div 5\)

These calculation errors lead to incorrect final answers and confusion about which form to use.

The Bottom Line:

This problem tests whether students truly understand what a y-intercept represents conceptually, then requires careful execution of multi-step algebra with fractions and negative numbers. The combination of conceptual understanding and computational precision makes it challenging for many students.