What is the equation of a line parallel to y = 4x - 1 through (1, 3)?

The equation of a line parallel to \( y = 4x - 1 \) through \( (1, 3) \) is \( y = 4x - 1 \).

To find the equation of a line parallel to a given line, we need to understand that parallel lines have the same gradient (or slope). The given line \( y = 4x - 1 \) has a gradient of 4. This is because the equation is in the form \( y = mx + c \), where \( m \) represents the gradient. Therefore, any line parallel to this one will also have a gradient of 4.

Next, we need to find the specific equation of the line that passes through the point \( (1, 3) \). We use the point-gradient form of a line equation, which is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the gradient. Substituting \( m = 4 \), \( x_1 = 1 \), and \( y_1 = 3 \) into the formula, we get:

\[ y - 3 = 4(x - 1) \]

Expanding and simplifying this equation:

\[ y - 3 = 4x - 4 \]
\[ y = 4x - 1 \]

So, the equation of the line parallel to \( y = 4x - 1 \) that passes through the point \( (1, 3) \) is indeed \( y = 4x - 1 \). This shows that the new line has the same gradient and passes through the given point, confirming it is parallel to the original line.