To solve the problem of finding a linear equation for the table, we will start by identifying the slope and y-intercept. The table shows a relationship between time (x) and distance (y):

Step 1: Find the slope (m)

The slope of a linear equation is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Using two points from the table, for example, $(2, 45)$ and $(7, 70)$:

$m = \frac{70 - 45}{7 - 2} = \frac{25}{5} = 5$

Step 2: Find the y-intercept (b)

The linear equation is in the form:

$y = mx + b$

Now that we know $m = 5$, substitute one of the points to find $b$. Let's use $(2, 45)$:

$45 = 5(2) + b \implies 45 = 10 + b \implies b = 35$

Step 3: Write the final equation

Thus, the linear equation is:

$y = 5x + 35$

This equation represents the relationship between time (x) and distance (y).

Do you have any questions or want more details on the steps? Here are 5 related questions you might find helpful:

1. How can you check if the linear equation is correct using other points from the table?
2. What does the slope represent in this context of time and distance?
3. How would the equation change if one of the values in the table were different?
4. Can you explain how to find the equation if the relationship wasn't linear?
5. What methods can be used to graph the linear equation?

Tip: Always double-check by plugging different points from the table into your equation to ensure consistency.