To find the linear equation that represents this table, let's analyze the relationship between $x$ and $y$.

Step 1: Identify the Pattern

Looking at the table:

• When $x = 3$, $y = 0$.
• When $x = 4$, $y = 1$.
• When $x = 5$, $y = 2$.
• When $x = 6$, $y = 3$.

The $y$-values increase by 1 as the $x$-values increase by 1. This indicates a slope of $m = 1$.

Step 2: Find the Equation

A linear equation has the form: $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

Since the slope $m = 1$, the equation becomes: $y = x + b$

Step 3: Solve for $b$

We can use the point $(3, 0)$ to find $b$: $0 = 3 + b$ $b = -3$

Final Equation

The equation that represents the table is: $y = x - 3$

Let me know if you need further clarification on any step!

Additional Questions:

1. What is the slope of a line and how is it calculated from a table?
2. How do you find the y-intercept when given points from a table?
3. Why does each point in a table follow a consistent pattern in a linear equation?
4. How would the equation change if the slope was different?
5. Can a table with non-linear data points still form a linear equation?

Tip: Always use at least two points to confirm the slope in a linear relationship from a table.