The table shows two values of x and their corresponding values of y. In the xy-plane, the graph of the...

1. TRANSLATE the problem information

• Given information:
  • Two points from a linear relationship: \((-18, -48)\) and \((7, 52)\)
  • The line passes through point \((\frac{4}{7}, \mathrm{a})\)
  • Need to find the value of 'a'

• What this tells us: We need to find the linear equation first, then substitute \(\mathrm{x} = \frac{4}{7}\) to find 'a'

2. INFER the solution strategy

• To find a point on a line, we need the equation of that line
• To find a linear equation, we need slope (m) and y-intercept (b)
• We can calculate slope from the two given points, then use either point to find the y-intercept

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m} = \frac{\mathrm{y_2} - \mathrm{y_1}}{\mathrm{x_2} - \mathrm{x_1}}\)
• \(\mathrm{m} = \frac{52 - (-48)}{7 - (-18)}\)
  \(= \frac{52 + 48}{25}\)
  \(= \frac{100}{25}\)
  \(= 4\)

4. SIMPLIFY to find the y-intercept

• Using \(\mathrm{y} = \mathrm{mx} + \mathrm{b}\) with point \((7, 52)\) and \(\mathrm{m} = 4\):
• \(52 = 4(7) + \mathrm{b}\)
• \(52 = 28 + \mathrm{b}\)
• \(\mathrm{b} = 24\)

5. TRANSLATE into the linear equation

• The equation is: \(\mathrm{y} = 4\mathrm{x} + 24\)

6. SIMPLIFY to find the value of 'a'

• Substitute \(\mathrm{x} = \frac{4}{7}\) into \(\mathrm{y} = 4\mathrm{x} + 24\):
• \(\mathrm{a} = 4(\frac{4}{7}) + 24\)
• \(\mathrm{a} = \frac{16}{7} + 24\)
• Convert 24 to sevenths: \(24 = \frac{168}{7}\)
• \(\mathrm{a} = \frac{16}{7} + \frac{168}{7}\)
  \(= \frac{184}{7}\)

Answer: \(\frac{184}{7}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students correctly find the slope and y-intercept but make arithmetic errors when adding fractions, particularly when converting 24 to sevenths or when adding \(\frac{16}{7} + \frac{168}{7}\).

Common mistakes include:

• Converting 24 incorrectly (getting \(\frac{144}{7}\) instead of \(\frac{168}{7}\))
• Adding the numerators incorrectly (16 + 168)
• Getting confused with fraction arithmetic entirely

This leads to confusion and often results in selecting an incorrect answer choice or guessing.

Second Most Common Error:

Missing INFER reasoning: Students try to work directly with the point \((\frac{4}{7}, \mathrm{a})\) without first establishing the linear equation. They might attempt to use ratios or proportions incorrectly, not recognizing that they need the complete linear equation first.

This causes them to get stuck early in the problem and resort to guessing among the answer choices.

The Bottom Line:

This problem requires systematic thinking - you must build the complete linear equation before you can find any specific point on it. The fraction arithmetic at the end, while straightforward, requires careful attention to avoid calculation errors.