The line passes through the point \( (4,-1) \) and is parallel to the line \( y=3 x+1 \)

To find the equation of the line that passes through the point \( (4,-1) \) and is parallel to the line \( y = 3x + 1 \), we need to use the slope of the given line. The slope of the line \( y = 3x + 1 \) is 3. Since parallel lines have the same slope, our new line will also have a slope of 3. We can use the point-slope form of a linear equation, which is given by: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the point the line passes through and \( m \) is the slope. Substituting in our values, we get: \[ y - (-1) = 3(x - 4) \] Simplifying this leads us to: \[ y + 1 = 3x - 12 \] Subtracting 1 from both sides gives: \[ y = 3x - 13 \] Voila! The equation of the line parallel to \( y = 3x + 1 \) and passing through the point \( (4, -1) \) is \( y = 3x - 13 \). Now, here's a bit more to spice things up! Imagine plotting this on a graph. You’ll find that all lines with the same slope of 3 look like they have a similar upward tilt, creating a cozy family of parallel lines. They never meet, kind of like two friends who take similar paths but never quite run into each other! It's also fun to visualize how many real-world scenarios can be represented by parallel lines. Think about streets in a grid pattern, rail tracks, or even layers of cake stacked perfectly. Each of these is a reminder of how structure and symmetry play a big role in our lives, echoing the beauty found in mathematics!