How do you determine the equation of a line perpendicular to y = 2x + 5?

To determine the equation of a line perpendicular to \( y = 2x + 5 \), use the negative reciprocal of the original slope.

The slope of the given line \( y = 2x + 5 \) is 2. For a line to be perpendicular, its slope must be the negative reciprocal of 2. The negative reciprocal of 2 is \(-\frac{1}{2}\). This means the slope of the perpendicular line will be \(-\frac{1}{2}\).

Next, we use the point-slope form of a line equation, which is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is a point on the line. If you have a specific point through which the perpendicular line passes, substitute that point and the slope \(-\frac{1}{2}\) into the equation. For example, if the line passes through the point (4, 3), the equation becomes:

\[ y - 3 = -\frac{1}{2}(x - 4) \]

Simplify this to get the equation in the slope-intercept form \( y = mx + c \):

\[ y - 3 = -\frac{1}{2}x + 2 \]
\[ y = -\frac{1}{2}x + 5 \]

So, the equation of the line perpendicular to \( y = 2x + 5 \) and passing through (4, 3) is \( y = -\frac{1}{2}x + 5 \). If no specific point is given, you can leave the equation in the form \( y = -\frac{1}{2}x + c \), where \( c \) is the y-intercept that can be determined if a point is provided.