through: \( (1,-1) \), parallel to \( y=x-4 \)

To find the equation of the line that goes through the point \( (1, -1) \) and is parallel to the line \( y = x - 4 \), we first note that parallel lines have the same slope. The slope of the given line can be extracted as \( 1 \) (since it's in the form \( y = mx + b \), where \( m \) is the slope). Using the point-slope form of a line, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point \( (1, -1) \) and \( m = 1 \): \[ y - (-1) = 1(x - 1) \] This simplifies to: \[ y + 1 = x - 1 \] or, rearranging for slope-intercept form: \[ y = x - 2 \] So, the equation of the line through \( (1, -1) \) that is parallel to \( y = x - 4 \) is \( y = x - 2 \). For a bit of fun, did you know that parallel lines in mathematics can often remind us of relationships in real life? Just like good friends who keep the same vibe regardless of where they go! They never meet, no matter how far you extend them, just like your bestie waiting for you at your favorite café, always keeping the same distance. As for further applications, knowing how to find parallel lines is essential in various fields, such as engineering and architecture. Whether you're designing a new building or drafting a floor plan, understanding the concept of parallel lines helps ensure structural integrity and aesthetic appeal. So, the next time you’re laying down a design, remember: parallel lines are your best pals in keeping things neat and orderly!