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To find the linear equation that matches the table provided, we need to determine the relationship between the values of \( x \) and \( y \).

Here is the table:

\[
\begin{array}{c|c}
x & y \\
\hline
4 & 1 \\
5 & 2 \\
6 & 3 \\
7 & 4 \\
\end{array}
\]

### Step 1: Determine the slope \( m \)

The slope \( m \) of the linear equation \( y = mx + b \) is calculated using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's use the first two points from the table: \( (4, 1) \) and \( (5, 2) \).

\[
m = \frac{2 - 1}{5 - 4} = \frac{1}{1} = 1
\]

So, the slope \( m = 1 \).

### Step 2: Find the y-intercept \( b \)

Now, we'll use one of the points to find the y-intercept \( b \). We can use the point \( (4, 1) \):

\[
y = mx + b
\]

Substituting the values:

\[
1 = 1(4) + b
\]

\[
1 = 4 + b
\]

\[
b = 1 - 4 = -3
\]

### Step 3: Write the equation

The linear equation that represents the table is:

\[
y = x - 3
\]

If you have any more questions or need further details, feel free to ask!

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Here are some related questions you might consider:

1. What is the significance of the slope in a linear equation?
2. How would the equation change if the y-intercept were positive?
3. How can you verify if a given point lies on the line \( y = x - 3 \)?
4. What would be the equation if the slope were doubled?
5. How can you graph the equation \( y = x - 3 \) using the slope and y-intercept?

**Tip:** Always double-check your calculations, especially when determining the slope and intercept, to ensure accuracy.

Write the linear equation that gives the rule for this table: (4, 1), (5, 2), (6, 3), (7, 4).

Learn how to derive the linear equation y = x - 3 from a table of values. This problem involves key algebraic concepts such as calculating the slope and finding the y-intercept. Suitable for students in Grades 7-8.

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