Math Problem Statement

Two lines are graphed on the coordinate plane.

Which description of the lines is TRUE?

A. The lines have the same slopes, but different -intercepts. The lines are a translation of each other by units.

B. The lines have the same slopes, but different -intercepts. The lines are a translation of each other by units.

C. The lines have different slopes, but the same -intercepts. The lines are a translation of each other by units.

D. The lines have different slopes, but the same -intercepts. The lines are a translation of each other by units.

Solution

To determine the correct answer, let's analyze the properties of the two lines based on their slopes and yy-intercepts.

  1. Same Slopes or Different Slopes:
    If the lines are parallel, they will have the same slope. If they are not parallel (meaning they intersect or are angled differently), they will have different slopes.

  2. Same yy-Intercepts or Different yy-Intercepts:
    If the lines cross the yy-axis at the same point, they have the same yy-intercept. Otherwise, they have different yy-intercepts.

  3. Translation
    Translating one line to overlap with another typically involves shifting vertically, horizontally, or both, depending on the direction and distance. This shift is often specified in units along the xx- or yy-axis.

From the options:

  • Option A suggests the lines are parallel (same slope) but have different yy-intercepts.
  • Option B also suggests parallel lines with different yy-intercepts but may imply a specific unit shift.
  • Options C and D indicate the lines have different slopes, which would mean they intersect or are angled differently.

To determine the best choice, consider that:

  1. If the lines appear to be parallel, A or B would be correct.
  2. If they appear to intersect or are angled, C or D would be correct.

Without viewing the exact graph here, a parallel interpretation aligns with options A and B.

If the graph clearly shows a parallel relationship, the correct answer would likely be B if the specific unit shift is confirmed in the context.

If you'd like, upload the graph image for a more precise determination!


Would you like further explanation on translations or more examples?

Here are five additional questions for practice:

  1. How do you determine the slope of a line from a graph?
  2. What is a yy-intercept, and why is it significant in graphing lines?
  3. What transformations exist beyond translations that can map one line onto another?
  4. How do parallel lines differ from perpendicular lines in terms of slopes?
  5. Can two lines with the same slope and different yy-intercepts ever intersect?

Tip: When identifying transformations on a graph, first check if the slopes match for a quick determination of whether lines are parallel!

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Parallel Lines
Translations

Formulas

Slope formula: (y2 - y1) / (x2 - x1)
Translation transformation formula

Theorems

Properties of parallel lines
Properties of translations in coordinate geometry

Suitable Grade Level

Grades 8-10