What is the equation of the line that passes through the point \((0, 5)\) and is parallel to the graph...

1. TRANSLATE the problem information

• Given information:
  • Line passes through point \((0, 5)\)
  • Line is parallel to \(\mathrm{y = 7x + 4}\)
• What this tells us: We need to find an equation in the form \(\mathrm{y = mx + b}\)

2. INFER what each piece of information provides

• The point \((0, 5)\): Since this is where the line crosses the y-axis, the y-intercept \(\mathrm{b = 5}\)
• "Parallel to \(\mathrm{y = 7x + 4}\)": Parallel lines have identical slopes, so our slope \(\mathrm{m = 7}\)

3. INFER the final equation structure

• We have slope \(\mathrm{m = 7}\) and y-intercept \(\mathrm{b = 5}\)
• Substituting into \(\mathrm{y = mx + b}\) gives us: \(\mathrm{y = 7x + 5}\)

Answer: B. \(\mathrm{y = 7x + 5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students mix up which information gives them slope versus y-intercept. They might think the point \((0, 5)\) means the slope is 5, or that being parallel to \(\mathrm{y = 7x + 4}\) means having the same y-intercept of 4.

This leads to equations like \(\mathrm{y = 5x + 7}\) (swapping the values) or \(\mathrm{y = 7x + 4}\) (copying the parallel line exactly).
This may lead them to select Choice D (\(\mathrm{y = 5x + 7}\)) or get confused between the remaining choices.

Second Most Common Error:

Incomplete conceptual knowledge about parallel lines: Students might remember that parallel lines are related but forget that they specifically have the same slope. They might think parallel lines have the same y-intercept instead.

This leads them to try to use the y-intercept from \(\mathrm{y = 7x + 4}\), creating confusion about which line property to match.
This may lead them to select Choice C (\(\mathrm{y = 7x}\)) by keeping the slope but getting confused about the y-intercept.

The Bottom Line:

This problem tests whether students can distinguish between information that determines slope (parallel line property) versus information that determines y-intercept (given point), then correctly combine them using slope-intercept form.