Which of the following equations defines a line that is parallel to the line \( y=-\frac{4}{3} x-4 \) and passes through the point \( (3,-1) \) ? A \( y=-\frac{4}{3} x-1 \) B \( y=-\frac{4}{3} x-3 \) C \( y=-\frac{4}{3} x+1 \) D \( y=-\frac{4}{3} x+3 \)

To find a line parallel to \( y = -\frac{4}{3} x - 4 \), we need to remember that parallel lines have the same slope. So, our new line will also have a slope of \( -\frac{4}{3} \). Now, we can use the point-slope form of a line, which starts with \( y - y_1 = m(x - x_1) \). Using the point \( (3, -1) \) and the slope \( -\frac{4}{3} \), we can derive the equation of the line. Plugging in the values, we have: \( y + 1 = -\frac{4}{3}(x - 3) \) This simplifies to: \( y + 1 = -\frac{4}{3}x + 4 \) Therefore, \( y = -\frac{4}{3}x + 3 \) Thus, the line that is parallel to the original and passes through the point \( (3, -1) \) is option D: \( y = -\frac{4}{3} x + 3 \). This hints at how geometry and algebra can work hand-in-hand! In real-world applications, the concept of parallel lines is essential, especially in fields like engineering, architecture, and even in art, where maintaining straight lines or levels is vital for structural integrity and aesthetic appeal. Additionally, it’s fascinating to note that many programming and computer graphics algorithms utilize linear equations to create realistic simulations and graphics. Everything we see on a screen, from the simplest 2D shapes to complex 3D models, often stems from these foundational concepts of lines and coordinates!