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To compute the **sum-of-squares error (SSE)** for the given set of data points \((4, 5), (5, 5), (7, 7)\) and the linear model \(y = x + 1\), follow these steps:

### 1. Predict \(y\) for each \(x\) using the model \(y = x + 1\).
For each \(x\) value from the data points, calculate the predicted \(y\) value.
- For \(x = 4\), predicted \(y = 4 + 1 = 5\).
- For \(x = 5\), predicted \(y = 5 + 1 = 6\).
- For \(x = 7\), predicted \(y = 7 + 1 = 8\).

### 2. Calculate the residual for each point.
The residual is the difference between the actual \(y\) value and the predicted \(y\) value.
- For \((4, 5)\): Residual = \(5 - 5 = 0\).
- For \((5, 5)\): Residual = \(5 - 6 = -1\).
- For \((7, 7)\): Residual = \(7 - 8 = -1\).

### 3. Square each residual.
- For \( (4, 5) \): \( 0^2 = 0 \).
- For \( (5, 5) \): \( (-1)^2 = 1 \).
- For \( (7, 7) \): \( (-1)^2 = 1 \).

### 4. Sum the squared residuals to get the SSE.
\[ \text{SSE} = 0 + 1 + 1 = 2. \]

### Final Answer:
The **sum-of-squares error (SSE)** is \( \boxed{2} \).

Would you like further clarification or additional details on this calculation? Here are some related questions you could explore:

1. How does the SSE help assess the accuracy of a model?
2. What is the difference between SSE and Mean Squared Error (MSE)?
3. Can SSE be negative, and why or why not?
4. How would the SSE change if the linear model was \(y = 2x + 1\)?
5. What are other metrics to evaluate the fit of a linear regression model?

**Tip**: The smaller the SSE, the closer the predicted values are to the actual values, indicating a better fit for the model.

Compute the sum-of-squares error (SSE) by hand for the given set of data and linear model: (4, 5), (5, 5), (7, 7); y = x + 1.

Learn how to compute the sum-of-squares error (SSE) for the data points (4, 5), (5, 5), and (7, 7) using the linear model y = x + 1. This problem covers key concepts such as residuals and SSE, making it suitable for students in Grades 10-12.

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