Find an equation of the line parallel to \( 9 x-y=2 \) and passing through \( (4,1) \). Express the equation in standard form. Which of the following is the equation of a line parallel to \( 9 x-y=2 \) and passing through \( (4,1) \) ?

To find a line parallel to \( 9x - y = 2 \), we first need to determine its slope by rewriting it in slope-intercept form (y = mx + b). Rearranging gives us \( y = 9x - 2 \), indicating that the slope (m) is 9. Since parallel lines share the same slope, the equation of our new line can be written as \( y - 1 = 9(x - 4) \). Simplifying this gives us \( y = 9x - 36 + 1 \), or \( y = 9x - 35\). To express this in standard form, we rearrange it: \( -9x + y = -35 \) or equivalently \( 9x - y = 35 \). So, the equation of the line parallel to \( 9x - y = 2 \) and passing through \( (4, 1) \) is \( 9x - y = 35 \). A fun fact about parallel lines is that no matter how far you extend them, they will never meet! Just like two best friends who always stick together but never cross paths. This concept is important in geometry and has practical applications, like when designing roads that need to maintain a consistent distance apart. Also, if you want to dive deeper into the world of lines, check out some resources on coordinate geometry! You’ll find that understanding the relationships between different types of lines can be used in everything from architecture to computer graphics, making it a versatile and essential topic.