What is the x-coordinate of the x-intercept of the graph of 7y/4 = -2x/3 + 14 in the xy-plane?Express your...

1. TRANSLATE the problem information

• Given: The equation \(\frac{7y}{4} = -\frac{2x}{3} + 14\)
• Need to find: x-coordinate of the x-intercept
• Key insight: x-intercept occurs where the graph crosses the x-axis, which means \(y = 0\)

2. TRANSLATE the intercept condition

• Set up the equation by substituting \(y = 0\):
  \(\frac{7(0)}{4} = -\frac{2x}{3} + 14\)
• This simplifies to: \(0 = -\frac{2x}{3} + 14\)

3. SIMPLIFY to isolate x

• Rearrange to get the x term by itself:
  \(\frac{2x}{3} = 14\)
• Multiply both sides by 3/2 to solve for x:
  \(x = 14 \times \frac{3}{2}\)
  \(x = \frac{42}{2}\)
  \(x = 21\)

Answer: 21

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not understanding that "x-intercept" means \(y = 0\)

Students might try to set \(x = 0\) instead, thinking they need to find where the graph crosses the y-axis. This leads them to solve \(\frac{7y}{4} = 14\), getting \(y = 8\), and incorrectly answering "8" or getting confused about what the question is actually asking for.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors with fraction multiplication

When solving \(x = 14 \times \frac{3}{2}\), students might incorrectly calculate this as \(x = 14 \times 3 \div 2 = 42 \div 2 = 21\)... wait, that's actually correct. More commonly, they might make errors like \(x = 14 \div 3 \times 2 = \frac{28}{3} \approx 9.33\), leading to an incorrect decimal answer.

The Bottom Line:

This problem tests whether students understand the fundamental concept of intercepts and can execute fraction arithmetic accurately. The key breakthrough is recognizing that "x-intercept" translates to "\(y = 0\)" - once that connection is made, the algebra becomes straightforward.