Let's solve each part step by step: --- ### **a. \( y = -3x + 1 \) and \( y = -3x + 11 \)** **Step 1: Compare the slopes and y-intercepts.** - Both equations are in the form \( y = mx + b \). - Both have the **same slope**: \( m = -3 \). - Different y-intercepts: \( 1 \) and \( 11 \). **Step 2: Nature of Intersection** - **Same slope, different y-intercepts** ⇒ **Parallel lines**. **Final Answer (a):** > **The lines are parallel and do not intersect.** --- ### **b. \( y = -x - 3 \) and \( x + y = -6 \)** **Step 1: Write both equations in slope-intercept form (\(y = mx + b\)):** Second equation: \[ x + y = -6 \implies y = -x - 6 \] So, the equations are: - \( y = -x - 3 \) - \( y = -x - 6 \) **Step 2: Compare slopes and y-intercepts.** - Both have slope \( m = -1 \). - Y-intercepts: \( -3 \) and \( -6 \). **Step 3: Nature of Intersection** - **Same slope, different y-intercepts** ⇒ **Parallel lines**. **Final Answer (b):** > **The lines are parallel and do not intersect.** --- ### **c. \( y = 8x + 2 \) and \( 12y = 96x + 24 \)** **Step 1: Write both equations in slope-intercept form (\(y = mx + b\)):** Second equation: \[ 12y = 96x + 24 \implies y = \frac{96x + 24}{12} = 8x + 2 \] So, both equations are: - \( y = 8x + 2 \) - \( y = 8x + 2 \) **Step 2: Compare slopes and y-intercepts.** - Both have the **same slope** and **same y-intercept**. **Step 3: Nature of Intersection** - **Same slope, same y-intercept** ⇒ **Coincident lines** (infinite solutions). **Final Answer (c):** > **The lines are coincident; they are the same line and have infinitely many points of intersection.** --- ## **Summary Table** | Part | Slopes | Y-intercepts | Nature of Intersection | |------|--------------|--------------------|--------------------------------------| | a | Same | Different | Parallel, do not intersect | | b | Same | Different | Parallel, do not intersect | | c | Same | Same | Coincident, infinitely many points |

a. The lines are parallel and do not intersect. b. The lines are parallel and do not intersect. c. The lines are coincident; they are the same line with infinitely many points of intersection.