 The equation of the line containing the points ​(negative 2​,negative 2​) and ​(2​,5​) is y equals 1.75 x plus 1.5. Compute the sum of the squared residuals of the given data set for this line.
The sum of the squared residuals for the line containing the points ​(negative 2​,negative 2​) and ​(2​,5​) is 
  
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To compute the sum of the squared residuals, we need to calculate the residual for each point and then square it before adding the results.

The **residual** for a point \((x_i, y_i)\) is the difference between the actual \(y_i\) value and the predicted \(y_i\) value from the line equation \(y = 1.75x + 1.5\).

Let's compute the residuals for both points:

1. **For the point (-2, -2):**

   - The actual \(y\) value is -2.
   - The predicted \(y\) value using \(y = 1.75x + 1.5\) is:
     \[
     y_{\text{predicted}} = 1.75(-2) + 1.5 = -3.5 + 1.5 = -2
     \]
   - The residual is:
     \[
     \text{residual} = y_{\text{actual}} - y_{\text{predicted}} = -2 - (-2) = 0
     \]
   - Squared residual:
     \[
     0^2 = 0
     \]

2. **For the point (2, 5):**

   - The actual \(y\) value is 5.
   - The predicted \(y\) value using \(y = 1.75x + 1.5\) is:
     \[
     y_{\text{predicted}} = 1.75(2) + 1.5 = 3.5 + 1.5 = 5
     \]
   - The residual is:
     \[
     \text{residual} = y_{\text{actual}} - y_{\text{predicted}} = 5 - 5 = 0
     \]
   - Squared residual:
     \[
     0^2 = 0
     \]

### Sum of squared residuals:
\[
\text{Sum} = 0 + 0 = 0
\]

Thus, the sum of the squared residuals is **0**.

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Would you like any additional details or clarifications?

Here are 5 related questions to expand on this topic:
1. What does a residual of 0 imply about the fit of the line to the points?
2. How would the sum of squared residuals change if the line did not pass through both points?
3. What is the general formula for calculating residuals for any given point?
4. How does least squares regression minimize the sum of squared residuals?
5. What is the significance of the slope and intercept in linear regression?

**Tip**:

The equation of the line containing the points (-2, -2) and (2, 5) is y equals 1.75x plus 1.5. Compute the sum of the squared residuals of the given data set for this line.

Learn how to compute the sum of squared residuals for the line y = 1.75x + 1.5 passing through the points (-2, -2) and (2, 5). This problem demonstrates key algebraic concepts like residuals and the sum of squared residuals, suitable for Grades 9-12.

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