Final answer:
The equations of lines that are perpendicular to the line y = 2x + 3 are y = -2x + 3 and y = -2x - 3, as their slopes are the negative reciprocals of the given line's slope.option b and d is correct answer.
Step-by-step explanation:
When discussing perpendicular lines, it's important to understand that the slope of one line is the negative reciprocal of the slope of the other line. If we are given a line with an equation, we can identify which equations represent lines perpendicular to it by comparing their slopes. For example, if the equation of the given line is y = 2x + 3, its slope is 2. This means, for a line to be perpendicular to it, the slope must be -1/2. The slope is represented by the coefficient of x in the equation y = mx + b, where m is the slope and b is the y-intercept.
Looking at the options:
- y = 2x + 3 (The given line)
- y = -2x + 3 (This is the correct answer since the slope is the negative reciprocal of 2, which is -1/2)
- y = 2x - 3 (This line has the same slope as the given line, so it's parallel, not perpendicular)
- y = -2x - 3 (Like the second option, this one also has a slope of -1/2, making it perpendicular to the given line)
Therefore, the equations of the lines that are perpendicular to the line y = 2x + 3 are b) y = -2x + 3 and d) y = -2x - 3.