What is the equation of a line parallel to y = 5x + 2 through (4, 1)?

The equation of a line parallel to \( y = 5x + 2 \) through \( (4, 1) \) is \( y = 5x - 19 \).

To understand why, let's start by recalling that parallel lines have the same gradient (slope). The given line \( y = 5x + 2 \) has a gradient of 5. Therefore, any line parallel to this one will also have a gradient of 5.

Next, we need to find the specific equation of the line that passes through the point \( (4, 1) \). We use the point-gradient form of a line equation, which is \( y - y_1 = m(x - x_1) \), where \( m \) is the gradient and \( (x_1, y_1) \) is a point on the line. Here, \( m = 5 \), \( x_1 = 4 \), and \( y_1 = 1 \).

Substituting these values into the point-gradient form, we get:
\[ y - 1 = 5(x - 4) \]

Now, we simplify this equation:
\[ y - 1 = 5x - 20 \]
\[ y = 5x - 19 \]

So, the equation of the line parallel to \( y = 5x + 2 \) that passes through the point \( (4, 1) \) is \( y = 5x - 19 \). This method ensures that the new line has the same gradient as the original line, making them parallel, and it passes through the given point.