What is the equation of the line that is perpendicular to \( y = 2x + 3 \) and passes through the origin?

To determine the equation of a line that is perpendicular to the line given by ( y = 2x + 3 ) and passes through the origin, it is important to understand the concept of slopes in relation to perpendicular lines. The slope of the given line is 2.

For two lines to be perpendicular, the product of their slopes must equal (-1). This means that if one line has a slope of ( m_1 ), the slope of a line that is perpendicular to it, ( m_2 ), can be found using the relationship:

[ m_1 \cdot m_2 = -1. ]

In our case, since the slope ( m_1 = 2 ), we can find the slope of the perpendicular line ( m_2 ) by solving:

[ 2 \cdot m_2 = -1 ]

This simplifies to:

[ m_2 = -\frac{1}{2}. ]

Once we have the slope of the perpendicular line, we can write the equation of the line in slope-intercept form, which is ( y = mx + b ). Since the line we are looking for passes through the origin, the y-intercept (