The line with the equation 4/5x + 1/3y = 1 is graphed in the xy-plane. What is the x-coordinate of...

1. TRANSLATE the problem information

• Given information:
  • Line equation: \(\frac{4}{5}\mathrm{x} + \frac{1}{3}\mathrm{y} = 1\)
  • Need to find: x-coordinate of the x-intercept

• What this tells us: At the x-intercept, the line crosses the x-axis, so \(\mathrm{y} = 0\)

2. SIMPLIFY by substituting y = 0

• Substitute \(\mathrm{y} = 0\) into the equation:
  \(\frac{4}{5}\mathrm{x} + \frac{1}{3}(0) = 1\)

• This simplifies to:
  \(\frac{4}{5}\mathrm{x} = 1\)

3. SIMPLIFY to solve for x

• To isolate x, multiply both sides by \(\frac{5}{4}\) (the reciprocal of \(\frac{4}{5}\)):
  \(\mathrm{x} = 1 \times \frac{5}{4} = \frac{5}{4}\)

• Convert to decimal: \(\mathrm{x} = 1.25\)

Answer: 1.25 (or 5/4)

Why Students Usually Falter on This Problem

Most Common Error Path:

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

Instead of setting \(\mathrm{y} = 0\), they might set \(\mathrm{x} = 0\), leading to:

\(\frac{4}{5}(0) + \frac{1}{3}\mathrm{y} = 1\)
\(\frac{1}{3}\mathrm{y} = 1\)
\(\mathrm{y} = 3\)

This would give them the y-intercept value of 3 instead of the x-intercept.

Second Most Common Error:

Poor SIMPLIFY execution: Students make computational errors when working with fractions.

Common mistakes include:

• Incorrectly multiplying by \(\frac{4}{5}\) instead of \(\frac{5}{4}\)
• Making arithmetic errors like getting \(\mathrm{x} = \frac{4}{5}\) instead of \(\mathrm{x} = \frac{5}{4}\)
• Converting \(\frac{5}{4}\) incorrectly to decimal form

This leads to various incorrect numerical answers and confusion about the final result.

The Bottom Line:

This problem tests whether students truly understand what an intercept represents geometrically and can accurately manipulate fractional coefficients in linear equations.