10. Select all the points that are on a line with slope 2 . that also contains the point \( (2,-1) \). (A.) \( (3,1) \) (B. \( (1,1) \) (C. \( (1,-3) \) (D. \( (4,0) \) (E. \( (6,7) \)

To determine which points lie on the line with a slope of 2 that also passes through the point \( (2, -1) \), follow these steps: ### 1. **Find the Equation of the Line** Given: - Slope (\( m \)) = 2 - Point on the line (\( x_1, y_1 \)) = \( (2, -1) \) Use the **point-slope form** of a linear equation: \[ y - y_1 = m(x - x_1) \] Plug in the known values: \[ y - (-1) = 2(x - 2) \\ y + 1 = 2x - 4 \\ y = 2x - 5 \] **Equation of the line:** \( y = 2x - 5 \) ### 2. **Check Each Point** Now, verify which of the given points satisfy the equation \( y = 2x - 5 \). - **Option A: \( (3, 1) \)** \[ y = 2(3) - 5 = 6 - 5 = 1 \\ \text{Given } y = 1 \Rightarrow \text{Matches} \] **Conclusion:** Lies on the line. - **Option B: \( (1, 1) \)** \[ y = 2(1) - 5 = 2 - 5 = -3 \\ \text{Given } y = 1 \Rightarrow \text{Does not match} \] **Conclusion:** Does **not** lie on the line. - **Option C: \( (1, -3) \)** \[ y = 2(1) - 5 = 2 - 5 = -3 \\ \text{Given } y = -3 \Rightarrow \text{Matches} \] **Conclusion:** Lies on the line. - **Option D: \( (4, 0) \)** \[ y = 2(4) - 5 = 8 - 5 = 3 \\ \text{Given } y = 0 \Rightarrow \text{Does not match} \] **Conclusion:** Does **not** lie on the line. - **Option E: \( (6, 7) \)** \[ y = 2(6) - 5 = 12 - 5 = 7 \\ \text{Given } y = 7 \Rightarrow \text{Matches} \] **Conclusion:** Lies on the line. ### 3. **Final Answer** **The points that lie on the line with slope 2 passing through \( (2, -1) \) are:** - **(A) \( (3, 1) \)** - **(C) \( (1, -3) \)** - **(E) \( (6, 7) \)** Options **A**, **C**, and **E** are correct.