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The table given presents pairs of \(X\) and \(Y\) coordinates. Let's denote the relationship between \(X\) and \(Y\) as a linear equation of the form \( Y = mX + b \), where \(m\) is the slope and \(b\) is the y-intercept. Our task is to find the missing \(X\) value corresponding to \(Y = -4\).

We have the following points:
\[
(6, 2), \quad (3, -1), \quad (?, -4)
\]

### Step 1: Find the slope
Using the first two points \((6, 2)\) and \((3, -1)\), we calculate the slope \(m\):
\[
m = \frac{Y_2 - Y_1}{X_2 - X_1} = \frac{-1 - 2}{3 - 6} = \frac{-3}{-3} = 1
\]
So, the slope \(m = 1\).

### Step 2: Find the y-intercept
Now, substitute one of the known points into the linear equation \( Y = mX + b \) to solve for \(b\). Using the point \((6, 2)\):
\[
2 = 1(6) + b \quad \Rightarrow \quad 2 = 6 + b \quad \Rightarrow \quad b = 2 - 6 = -4
\]

Thus, the equation of the line is:
\[
Y = X - 4
\]

### Step 3: Solve for the missing \(X\)
Now, we substitute \(Y = -4\) into the equation to find the missing \(X\):
\[
-4 = X - 4 \quad \Rightarrow \quad X = 0
\]

### Conclusion:
The missing \(X\) coordinate is \(0\).

Let me know if you'd like more details, and feel free to ask questions! Here are some related questions:

1. How can you derive the slope formula between two points?
2. What are the general steps to finding the equation of a line?
3. Can you solve for other missing values if a different relationship (like quadratic) exists between \(X\) and \(Y\)?
4. How would the solution change if the points were non-linear?
5. What other methods can you use to determine missing coordinates besides linear equations?

**Tip:** Always check if the relationship between variables is linear by calculating the slope between multiple points. If the slope remains consistent, the relationship is linear.

Discover how to solve for missing coordinates using linear equations with a clear, detailed example. This tutorial explains how to determine the slope and y-intercept and apply these to find the missing X or Y values, ideal for students in Grades 7-10.

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