Assignment Aline passes through \( (0,-2) \) and \( (4,6) \) Find the equation of a line parallel to this Nine passing through \( (-3,-1) \)

To find the equation of the line parallel to the one passing through points \( (0,-2) \) and \( (4,6) \), we first need to determine the slope of the original line. The slope \( m \) can be calculated using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using points \( (0,-2) \) and \( (4,6) \): \[ m = \frac{6 - (-2)}{4 - 0} = \frac{8}{4} = 2 \] Since parallel lines have equal slopes, the slope of our new line will also be 2. Now we can use the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (-3, -1) \): \[ y - (-1) = 2(x - (-3)) \] This simplifies to: \[ y + 1 = 2(x + 3) \] \[ y + 1 = 2x + 6 \] \[ y = 2x + 5 \] So, the equation of the line parallel to the original and passing through \( (-3,-1) \) is: $$ y = 2x + 5 $$ In this case, the equation neatly describes a straight line sloping upward, illustrating the beauty of linear relationships! The slope of 2 means that for every step you take to the right, you move up two steps, making it quite steep and energetic! If you need to visualize or graph this line, consider plotting the point \( (-3, -1) \) and then use the slope to find other points. Moving right by 1 (to -2) causes you to move up by 2 to 1, and continuing this will create a straight line that slices through the coordinate plane! Each point communicates a consistent increase, an exciting dance of numbers!