Short Answer 
Understanding linear relationships involves identifying whether they are proportional (passing through the origin) or non-proportional (not passing through the origin). The equation of a line, y = ax + b, helps distinguish between these types based on the value of b, with real-world examples like cell phone pricing illustrating non-proportional relationships.
Step 1: Identify the Type of Linear Relationship
There are two main types of linear relationships: proportional and non-proportional. A proportional relationship is characterized by a line that crosses through the origin (0,0), showing that the output is directly linked to the input. In contrast, a non-proportional relationship does not pass through the origin, indicating that there is a constant value (b) affecting the output.
Step 2: Understand the Equation of a Line
The general equation for a line is represented as y = ax + b, where a is the slope and b is the y-intercept. If b equals zero, the line will cross through the origin, indicating a proportional relationship. If b is greater than zero, the line intercepts the y-axis at that point, leading to a non-proportional relationship.
Step 3: Analyze Real-World Examples
Consider the case of a cell phone company. If they charge $19.95 per month for the first 2GB, and then $5 for each additional gig, this represents a non-proportional relationship because the cost never reaches zero, and the line does not intersect the origin. Similarly, a flat monthly fee of $29.95 also illustrates a non-proportional trend, highlighting the importance of understanding these types of relationships in practical scenarios.