The graph shows the relationship between the number of shares of stock from Company A, x, and the number of...

1. TRANSLATE the graph information

Looking at the graph, I need to identify clear points that the line passes through:

• The line crosses the y-axis (where \(\mathrm{x = 0}\)) at the point \((0, 40)\)
• The line crosses the x-axis (where \(\mathrm{y = 0}\)) at the point \((60, 0)\)

These are the two clearest points to read from the graph because they're at the intercepts.

2. INFER the solution strategy

Here's the key insight: If points lie on a line, they must satisfy the equation of that line.

This means both \((0, 40)\) and \((60, 0)\) should make the correct equation true when I substitute their coordinates.

My strategy: Test each answer choice by substituting both points and see which equation works for both.

3. SIMPLIFY by testing each equation systematically

Testing Choice A: \(\mathrm{y = 8x + 12}\)

• Substitute point \((0, 40)\): Does \(40 = 8(0) + 12\)?

\(40 = 0 + 12 = 12\) ✗

This doesn't work, so Choice A is eliminated.

Testing Choice B: \(\mathrm{8x + 12y = 480}\)

• Substitute point \((0, 40)\): Does \(8(0) + 12(40) = 480\)?

\(0 + 480 = 480\) ✓

• Substitute point \((60, 0)\): Does \(8(60) + 12(0) = 480\)?

\(480 + 0 = 480\) ✓

Both points work perfectly! This is very likely our answer, but let's check the others to be sure.

Testing Choice C: \(\mathrm{y = 12x + 8}\)

• Substitute point \((0, 40)\): Does \(40 = 12(0) + 8\)?

\(40 = 0 + 8 = 8\) ✗

This doesn't work, so Choice C is eliminated.

Testing Choice D: \(\mathrm{12x + 8y = 480}\)

• Substitute point \((0, 40)\): Does \(12(0) + 8(40) = 480\)?

\(0 + 320 = 320 \neq 480\) ✗

This doesn't work either.

Answer: B. \(\mathrm{8x + 12y = 480}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misreading the coordinates from the graph.

Students might misread the y-intercept as \((0, 30)\) or \((0, 50)\) instead of \((0, 40)\), or misread the x-intercept as \((50, 0)\) instead of \((60, 0)\). Even a small error in reading the graph means all subsequent testing will give incorrect results. For example, if a student reads \((0, 30)\) and tests it in Choice B: \(8(0) + 12(30) = 360 \neq 480\), they would incorrectly eliminate the right answer. This leads to confusion and guessing among the wrong choices.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors during substitution, particularly when multiplying.

Students might calculate \(12 \times 40\) incorrectly (getting 400 or 440 instead of 480), or miscalculate \(8 \times 60\). These errors cause them to reject the correct equation and potentially select Choice D (\(\mathrm{12x + 8y = 480}\)) if they confuse the coefficients or make compensating errors that seem to work.

Third Most Common Error:

Weak INFER skill: Not recognizing that testing answer choices with known points is an efficient strategy, or thinking that standard form equations are "wrong" because they don't look like \(\mathrm{y = mx + b}\).

Some students only check Choices A and C because they're in slope-intercept form and look "simpler," dismissing Choices B and D without testing them. This causes them to get stuck choosing between two wrong answers and leads to guessing.

The Bottom Line:

This problem requires careful reading of coordinates from a visual representation and systematic testing. Success depends on precision in both reading the graph and performing substitution arithmetic—small errors at either stage derail the entire solution.