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The problem asks to write an equation for a tile pattern, and the hint suggests creating a table that relates the figure number (\(x\)) to the number of squares (\(y\)) in the pattern.

Let's first observe the given figures:

- **Figure 2** has 5 tiles.
- **Figure 3** has 7 tiles.
- **Figure 4** has 9 tiles.

From this, we can create the following table for the number of squares (\(y\)) in terms of the figure number (\(x\)):

| \(x\) (Figure Number) | \(y\) (Number of Squares) |
|-----------------------|---------------------------|
| 2                     | 5                         |
| 3                     | 7                         |
| 4                     | 9                         |

### Step 1: Identifying the pattern
The number of squares increases by 2 with each increase in the figure number. This suggests a linear relationship.

### Step 2: Writing the equation
The general form of a linear equation is:

\[
y = mx + b
\]

Where:
- \(m\) is the slope (the rate of change).
- \(b\) is the y-intercept (the value of \(y\) when \(x = 0\)).

#### Finding the slope (\(m\)):
The slope is the rate of change of \(y\) with respect to \(x\):

\[
m = \frac{\Delta y}{\Delta x} = \frac{7 - 5}{3 - 2} = 2
\]

So, the slope \(m = 2\).

#### Finding the y-intercept (\(b\)):
We can use one of the points to solve for \(b\). Using the point \((x = 2, y = 5)\):

\[
5 = 2(2) + b \quad \Rightarrow \quad 5 = 4 + b \quad \Rightarrow \quad b = 1
\]

### Final equation:
The equation that represents the pattern is:

\[
y = 2x + 1
\]

This means that for any figure number \(x\), the number of squares \(y\) can be found using this equation.

Would you like further details on this solution or have any questions?

Here are 5 related questions you might find interesting:
1. What would the number of squares be for Figure 5?
2. How would the equation change if the pattern added 3 squares instead of 2 between each figure?
3. Can you derive an equation for a different pattern based on a similar linear relationship?
4. What does the slope of 2 represent in the context of this pattern?
5. How would the graph of this equation look?

**Tip**: Whenever you see a consistent change in values, it often suggests a linear relationship.

Write the equation for the following tile pattern. Make a table of data, relating the figure number (as x) and the number of squares (y).

Learn how to write a linear equation for a tile pattern by observing how the number of squares changes with each figure. Understand how to create a table of values and use the slope-intercept form y = mx + b to solve the problem. Suitable for Grades 6-8.

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