2).How does a pattern of change between two variables (such as distance (d) and time ( t )) in a linear relationship show up in a table? (9pts) For example ........

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In a linear relationship between two variables, such as distance (d) and time (t), the pattern of change is characterized by a constant rate of change. This consistency is clearly reflected in a table of values for these variables. Here's how this pattern typically manifests:

1. **Constant Differences**: As one variable increases by equal increments, the other variable increases (or decreases) by the same fixed amount. This means that the change in distance per unit time remains constant. For example:

| Time (t) | Distance (d) |
   |----------|--------------|
   | 1        | 5            |
   | 2        | 10           |
   | 3        | 15           |
   | 4        | 20           |

In this table, for each increase of 1 unit in time, distance increases by 5 units. This consistent difference (Δd/Δt = 5) indicates a linear relationship where $$ d = 5t $$.

2. **Proportional Relationship**: The table may also show that the variables are directly proportional, meaning they increase at the same rate. The ratio of distance to time ($$ d/t $$) remains constant across all entries. Using the previous example, $$ d/t = 5 $$ for all rows, reinforcing the linearity.

3. **Graphical Representation Alignment**: If you were to plot these table values on a graph, the points would align along a straight line. This visual confirmation supports the linear pattern observed in the table. The slope of this line corresponds to the constant rate of change identified in the table.

**Example Table Showing Linear Relationship:**

| Time (t) | Distance (d) |
|----------|--------------|
| 0        | 0            |
| 1        | 5            |
| 2        | 10           |
| 3        | 15           |
| 4        | 20           |
| 5        | 25           |

In this example:

- The change in distance ($$ \Delta d $$) is consistently 5 units for every 1-unit change in time ($$ \Delta t $$).
- The relationship can be described by the equation $$ d = 5t $$, illustrating a straight-line, linear relationship.
- The table clearly displays the linear pattern through steady, predictable increments in distance as time progresses.

This consistent and predictable pattern in the table signifies a linear relationship between the two variables, making it easy to predict one variable based on the other.
In a linear relationship, the table shows that as one variable increases by equal amounts, the other variable increases by the same fixed amount. For example, if time increases by 1 unit, distance increases by 5 units consistently. This steady pattern means the relationship can be represented by a straight line on a graph, confirming the linear nature of the relationship.

2).How does a pattern of change between two variables (such as distance (d) and time ( t )) in a linear relationship show up in a table? (9pts) For example ........

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