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The table provides values for \(x\) and corresponding values for \(y\). The task is to find which equation fits all pairs of values.

The pairs are:

- When \(x = 1\), \(y = 0\)
- When \(x = 2\), \(y = -2\)
- When \(x = 3\), \(y = -4\)
- When \(x = 4\), \(y = -6\)

We can observe that as \(x\) increases by 1, \(y\) decreases by 2. This indicates that the slope of the line is \(-2\).

Now, let's check the provided options:

1. **Option A: \( y = -2x + 2 \)**  
   Substituting \(x = 1\):  
   \( y = -2(1) + 2 = 0 \), which is correct.  
   Let's try \(x = 2\):  
   \( y = -2(2) + 2 = -2 \), correct again.  
   Similarly, checking for \(x = 3\) and \(x = 4\) also satisfies the equation.  
   So, this seems to be the correct equation.

2. **Option B: \( y = x - 1 \)**  
   For \(x = 1\):  
   \( y = 1 - 1 = 0 \), correct.  
   But for \(x = 2\):  
   \( y = 2 - 1 = 1 \), which is incorrect.  
   So, this is not the correct option.

3. **Option C: \( y = 2x - 2 \)**  
   For \(x = 1\):  
   \( y = 2(1) - 2 = 0 \), correct.  
   But for \(x = 2\):  
   \( y = 2(2) - 2 = 2 \), which is incorrect.  
   Hence, not the correct option.

4. **Option D: \( y = -2x - 1 \)**  
   For \(x = 1\):  
   \( y = -2(1) - 1 = -3 \), which is incorrect.  
   So, this is not the correct option.

**Conclusion: Option A (\(y = -2x + 2\)) is the correct answer.**

Would you like further details or have any questions?

### Related Questions:
1. How do we find the slope from a table of values?
2. Can this process be applied to non-linear equations?
3. What is the significance of the \(y\)-intercept in these equations?
4. How can we graphically verify this equation?
5. What happens when the slope is positive instead of negative?

**Tip**: Always verify your equation by substituting multiple points from the table to ensure accuracy.

Which is true for all pairs of values in this table? The table shows values for x = {1, 2, 3, 4} and y = {0, -2, -4, -6}.

Learn how to determine the linear equation that fits a given set of values from a table, using the equation y = -2x + 2. This problem involves identifying the slope and y-intercept from data points and is suitable for students in Grades 7-9.

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