The table shows the linear relationship between the number of hours a smartphone has been used continuously, h, and the...

1. TRANSLATE the problem information

• Given information:
  • Table with three coordinate pairs: \((2, 86)\), \((4, 72)\), \((7, 51)\)
  • Need to find equation relating h (hours) and b (battery percentage)
  • Answer choices are in the form: coefficient×h ± b = constant

2. INFER the solution approach

• Since this is a linear relationship, I need to find the slope and y-intercept
• Strategy: Use slope formula with two points, then find y-intercept using one point
• Finally, rearrange the equation to match the answer format

3. SIMPLIFY to find the slope

• Using points \((2, 86)\) and \((4, 72)\):
• \(\mathrm{slope} = \frac{72 - 86}{4 - 2} = \frac{-14}{2} = -7\)

4. SIMPLIFY to find the y-intercept

• Using point \((2, 86)\) and slope = -7:
• \(\mathrm{b} = -7\mathrm{h} + \mathrm{c}\)
• \(86 = -7(2) + \mathrm{c}\)
• \(86 = -14 + \mathrm{c}\)
• \(\mathrm{c} = 100\)

5. SIMPLIFY to write and rearrange the equation

• Linear equation: \(\mathrm{b} = -7\mathrm{h} + 100\)
• Rearrange to match answer format: \(7\mathrm{h} + \mathrm{b} = 100\)

6. Verify with the third point

• Check with \((7, 51)\): \(7(7) + 51 = 49 + 51 = 100\) ✓

Answer: A \((7\mathrm{h} + \mathrm{b} = 100)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when calculating the slope or rearranging equations.

For example, they might calculate slope as +7 instead of -7, leading to the equation \(\mathrm{b} = 7\mathrm{h} + \mathrm{c}\). After finding \(\mathrm{c} = 72\), they get \(\mathrm{b} = 7\mathrm{h} + 72\), which rearranges to \(7\mathrm{h} - \mathrm{b} = -72\). Since this doesn't match any choice exactly, they might incorrectly select Choice C \((7\mathrm{h} + \mathrm{b} = -100)\) thinking the signs are related.

Second Most Common Error:

Poor INFER reasoning about equation format: Students find the correct slope and y-intercept but struggle with rearranging \(\mathrm{b} = -7\mathrm{h} + 100\) into the standard form.

They might incorrectly rearrange as \(7\mathrm{h} - \mathrm{b} = 100\) instead of \(7\mathrm{h} + \mathrm{b} = 100\), leading them to select Choice D \((7\mathrm{h} - \mathrm{b} = 100)\).

The Bottom Line:

This problem tests whether students can systematically work through finding a linear equation AND properly manipulate algebraic expressions to match a given format - both skills are essential for success.