if this is the eaquation of a line y=2x+3 what is the parallel line and perpendicular line?
To find the parallel and perpendicular lines to the equation y = 2x + 3, we need to understand the properties of slope in linear equations.
In a linear equation of the form y = mx + b, the slope (m) represents the rate of change of the line. To find a parallel line, we need to have the same slope, whereas for a perpendicular line, we need to find the negative reciprocal slope.
1. Parallel Line:
Since the given equation is y = 2x + 3, the slope of this line is 2. For a parallel line, we can choose any equation with the same slope. Let's say we want to find the equation of a parallel line passing through the point (4, 7).
To find the equation of a parallel line, you can use the point-slope form of a linear equation. The point-slope form is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope of the line.
Using the point (4, 7) and the slope of 2, we get:
y - 7 = 2(x - 4)
y - 7 = 2x - 8
y = 2x - 1
Therefore, the equation of a parallel line to y = 2x + 3 is y = 2x - 1.
2. Perpendicular Line:
To find the equation of a perpendicular line, we need to find the negative reciprocal slope of the given line. For y = 2x + 3, the slope is 2. The negative reciprocal of 2 is -1/2.
Again, we can use the point-slope form to find the equation of a perpendicular line. Let's use the same point (4, 7) to find the equation of a perpendicular line.
Using the point (4, 7) and the slope of -1/2:
y - 7 = (-1/2)(x - 4)
y - 7 = (-1/2)x + 2
y = (-1/2)x + 9
Therefore, the equation of a perpendicular line to y = 2x + 3 is y = (-1/2)x + 9.