The table shows three values of x and their corresponding values of y. There is a linear relationship between x...

1. TRANSLATE the problem information

• Given: A table with three (x,y) pairs showing a linear relationship
• Need: An equation in the form \(\mathrm{y = mx + b}\)
• Available choices: Four different linear equations

2. INFER the most efficient approach

• Since we need \(\mathrm{y = mx + b}\), we need to find m (slope) and b (y-intercept)
• Key insight: When \(\mathrm{x = 0}\), the equation becomes \(\mathrm{y = b}\), so we can read the y-intercept directly
• From the table: when \(\mathrm{x = 0}\), \(\mathrm{y = 18}\), so \(\mathrm{b = 18}\)

3. SIMPLIFY to find the slope

• Now we know: \(\mathrm{y = mx + 18}\)
• Use another point to find m. From the table: when \(\mathrm{x = 1}\), \(\mathrm{y = 13}\)
• Substitute: \(\mathrm{13 = m(1) + 18}\)
• Solve:
  \(\mathrm{13 = m + 18}\)
  \(\mathrm{m = 13 - 18 = -5}\)

4. INFER the complete equation

• With \(\mathrm{m = -5}\) and \(\mathrm{b = 18}\): \(\mathrm{y = -5x + 18}\)
• SIMPLIFY verification with the third point:
  When \(\mathrm{x = 2}\):
  \(\mathrm{y = -5(2) + 18}\)
  \(\mathrm{y = -10 + 18}\)
  \(\mathrm{y = 8}\) ✓

Answer: D. \(\mathrm{y = -5x + 18}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret which values represent the slope and y-intercept, often thinking the y-intercept should be 13 (the middle y-value) rather than recognizing that \(\mathrm{x = 0}\) directly gives the y-intercept of 18.

This confusion about the meaning of y-intercept leads them to select Choice C (\(\mathrm{y = -5x + 13}\)) where they correctly find the slope but use the wrong y-intercept.

Second Most Common Error:

Poor INFER reasoning: Students fail to recognize that the y-values are decreasing as x increases, meaning the slope must be negative. They might calculate \(\mathrm{|13 - 18| = 5}\) but forget the negative sign.

This leads them to select Choice A (\(\mathrm{y = 18x + 13}\)) or Choice B (\(\mathrm{y = 13x + 18}\)) where they mix up the coefficients and miss the negative slope.

The Bottom Line:

This problem tests whether students truly understand what y-intercept means (the value when \(\mathrm{x = 0}\)) and can systematically work with the linear equation form rather than just memorizing formulas.