The relationship between two variables, x and y, is linear. For every increase in the value of x by 1,...

1. TRANSLATE the problem information

• Given information:
  • The relationship is linear
  • For every increase in x by 1, y increases by 8
  • When x = 2, y = 18

• What this tells us:
  • The slope is 8 (since y changes by 8 when x changes by 1)
  • We have the point (2, 18) on our line

2. INFER the approach

• Since we need a linear equation, we'll use slope-intercept form: \(\mathrm{y = mx + b}\)
• We know the slope (\(\mathrm{m = 8}\)), so we need to find the y-intercept (b)
• We can use the given point to find b

3. Set up the equation with known slope

• With slope \(\mathrm{m = 8}\): \(\mathrm{y = 8x + b}\)
• Now we need to find b

4. SIMPLIFY to find the y-intercept

• Substitute the point (2, 18):

\(\mathrm{18 = 8(2) + b}\)
\(\mathrm{18 = 16 + b}\)
\(\mathrm{b = 2}\)

5. Write the final equation

• \(\mathrm{y = 8x + 2}\)

Answer: C. \(\mathrm{y = 8x + 2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the slope with the y-intercept values from the problem statement.

They might think "y increases by 8" means \(\mathrm{b = 8}\), or "when x = 2, y = 18" means the slope is \(\mathrm{18/2 = 9}\). They might also think the equation should be \(\mathrm{y = 2x + 18}\) because those are the specific values given.

This may lead them to select Choice A (\(\mathrm{y = 2x + 18}\)) or Choice B (\(\mathrm{y = 2x + 8}\)).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify \(\mathrm{slope = 8}\) and set up \(\mathrm{y = 8x + b}\), but make an arithmetic error when solving \(\mathrm{18 = 16 + b}\).

They might incorrectly calculate \(\mathrm{b = 18 - 16 = 2}\), but then write it incorrectly in the final equation, or make sign errors during substitution.

This causes confusion and may lead to guessing among the remaining choices.

The Bottom Line:

Linear relationship problems require careful translation of verbal descriptions into mathematical relationships. The key insight is recognizing that "rate of change" language directly gives you the slope, not other parts of the equation.