The graph models the relationship between the number of T-shirts, x, and the number of sweatshirts, y, that Kira can...

1. TRANSLATE the graph information

Looking at the graph, we need to identify two clear points on the line:

• Point 1 (y-intercept): Where the line crosses the y-axis at \(x = 0\)
  The line crosses at \(y = 35\), so the point is \((0, 35)\)
• Point 2 (x-intercept): Where the line crosses the x-axis at \(y = 0\)
  The line crosses at \(x = 90\), so the point is \((90, 0)\)

2. INFER what equation form to use first

Since we have two points, we can find the slope and then use slope-intercept form \((y = mx + b)\). The y-intercept is already visible as 35.

3. SIMPLIFY to find the slope

Using the slope formula with \((0, 35)\) and \((90, 0)\):

\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

\(m = \frac{0 - 35}{90 - 0}\)

\(m = \frac{-35}{90}\)

Reduce the fraction: \(\frac{-35}{90} = \frac{-7}{18}\)

4. Write the equation in slope-intercept form

We have:

• Slope: \(m = \frac{-7}{18}\)
• Y-intercept: \(b = 35\)

Therefore: \(y = \frac{-7}{18}x + 35\)

5. INFER that we need to convert forms

Looking at the answer choices, none are in slope-intercept form. Choices B and D are in standard form \((Ax + By = C)\). We need to convert our equation.

6. SIMPLIFY by converting to standard form

Starting with: \(y = \frac{-7}{18}x + 35\)

Multiply both sides by 18 to eliminate the fraction:

\(18y = -7x + 630\)

Add \(7x\) to both sides to get standard form:

\(7x + 18y = 630\)

7. TRANSLATE back to verify

Check if our equation works with the original points:

• Point \((0, 35)\): \(7(0) + 18(35) = 0 + 630 = 630\) ✓
• Point \((90, 0)\): \(7(90) + 18(0) = 630 + 0 = 630\) ✓

Answer: B. \(7x + 18y = 630\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Process Skill Error (Weak TRANSLATE): Misreading the intercepts from the graph

Students may misidentify the coordinates, particularly:

• Reading the y-intercept as \((0, 30)\) instead of \((0, 35)\)
• Reading the x-intercept as \((80, 0)\) or \((100, 0)\) instead of \((90, 0)\)

This leads to calculating an incorrect slope and equation that doesn't match any answer choice, causing confusion and guessing.

Second Most Common Error:

Process Skill Error (Inadequate SIMPLIFY execution): Stopping at slope-intercept form without converting

Students correctly derive \(y = \frac{-7}{18}x + 35\) but don't realize they need to convert this to standard form to match the answer choices. They look at the options, don't see their equation, and either:

• Guess randomly among the choices
• Try to "force fit" by assuming Choice A or C might work since they're simpler
• Make arithmetic errors while trying to verify choices

This leads to confusion and guessing, or potentially selecting Choice A \((y = 7x + 18)\) because it has the numbers 7 and 18 from their correct slope.

Third Common Error:

Process Skill Error (Weak SIMPLIFY): Making sign errors during conversion

Students may correctly identify the slope as \(\frac{-7}{18}\) but make errors when converting to standard form:

• Forgetting to add \(7x\) to both sides (leaving it as \(18y = -7x + 630\))
• Placing coefficients on the wrong variables
• Getting \(18x + 7y = 630\) instead of \(7x + 18y = 630\)

This may lead them to select Choice D \((18x + 7y = 630)\), which has the coefficients swapped.

The Bottom Line:

This problem tests multiple skills in sequence: accurate graph reading, calculating slope, understanding equation forms, and algebraic manipulation. Students often succeed at one or two stages but fail to complete the full process, particularly the conversion from slope-intercept to standard form.