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

1. TRANSLATE the problem information

• Given information:
  • Table with four (x, y) coordinate pairs
  • Linear relationship exists between x and y
  • Need to match this relationship to one of four equations in standard form

• What this tells us: We need to find the equation of the line that passes through all four points

2. INFER the solution strategy

• To find a linear equation, we need the slope and y-intercept
• We can use any two points from the table to find the slope
• Once we have the slope, we can substitute any point to find the y-intercept
• Then we'll need to convert our equation to match the format of the answer choices

3. SIMPLIFY to find the slope

• Using points \((-6, 65)\) and \((6, 29)\):
  \(\mathrm{m = \frac{29 - 65}{6 - (-6)}}\)
  \(\mathrm{= \frac{-36}{12}}\)
  \(\mathrm{= -3}\)

4. SIMPLIFY to find the y-intercept

• Using \(\mathrm{y = mx + b}\) with point \((-6, 65)\):
  \(\mathrm{65 = -3(-6) + b}\)
  \(\mathrm{65 = 18 + b}\)
  \(\mathrm{b = 47}\)

5. INFER the equation conversion needed

• Our equation: \(\mathrm{y = -3x + 47}\)
• Answer choices are in standard form \(\mathrm{ax + by = c}\)
• Need to rearrange: \(\mathrm{3x + y = 47}\)
• Looking at the coefficients in answer choices, multiply by 3: \(\mathrm{9x + 3y = 141}\)

Answer: A. \(\mathrm{9x + 3y = 141}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make calculation errors when finding the slope or when converting between equation forms.

For example, they might calculate the slope as \(\mathrm{\frac{-36}{12} = 3}\) instead of \(\mathrm{-3}\), or make sign errors when rearranging the equation. These algebraic mistakes lead to equations that don't match any of the answer choices, causing confusion and random guessing.

Second Most Common Error:

Poor INFER reasoning about equation forms: Students find the correct slope-intercept form \(\mathrm{y = -3x + 47}\) but don't recognize how to convert it to match the standard form given in the answer choices.

They may try to force their equation into one of the choices without proper algebraic manipulation, or they may not realize they need to multiply both sides by 3 to get the coefficients that appear in choice A. This may lead them to select Choice C \(\mathrm{(3x + 9y = 141)}\) by incorrectly distributing the multiplication.

The Bottom Line:

This problem tests whether students can systematically work through the complete process from data table to final equation form, requiring both computational accuracy and strategic thinking about equation manipulation.