Find the equation of the line perpendicular to y=2x+6 and passing through (4,5)
I know that perpendicular lines are negative reciprocals of each other in terms of slope, so the slope will be -1/2 (-1/2 times 2 = -1)
To find the equation of a line that is perpendicular to the given line y = 2x + 6, we need to first determine the slope of the given line.
The given line, y = 2x + 6, is in slope-intercept form y = mx + b, where m is the slope of the line. In this case, the slope m is 2.
For a line to be perpendicular, the slope of the perpendicular line is the negative reciprocal of the original slope. So, the slope of the perpendicular line will be -1/2.
Now that we have the slope of the perpendicular line, we can use the point-slope form of a line to find the equation.
The point-slope form of a line is given by the equation y - y1 = m(x - x1), where m is the slope and (x1, y1) are the coordinates of a point on the line.
Plugging in the values from the point (4,5) and the negative reciprocal slope -1/2, we get:
y - 5 = (-1/2)(x - 4)
Simplifying this equation, we have:
y - 5 = (-1/2)x + 2
Adding 5 to both sides of the equation, we get:
y = (-1/2)x + 7
Therefore, the equation of the line perpendicular to y = 2x + 6 and passing through (4,5) is y = (-1/2)x + 7.
To find the equation of a line that is perpendicular to a given line, we need to determine the slope of the given line and then calculate the negative reciprocal of that slope.
The given line has an equation y = 2x + 6. Comparing this equation to the standard form y = mx + b, we can see that the slope of the given line is 2.
To find the slope of the line perpendicular to the given line, we take the negative reciprocal of 2. The negative reciprocal of 2 is -1/2.
Now that we have the slope (-1/2) and a point on the line (4, 5), we can use the point-slope form of a line to find the equation.
The point-slope form of a line is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope.
Using (x1, y1) = (4, 5) and m = -1/2, we substitute the values into the formula:
y - 5 = (-1/2)(x - 4)
We can simplify this equation:
y - 5 = (-1/2)x + 2
Now, we can rearrange the equation to get it in the slope-intercept form, y = mx + b:
y = (-1/2)x + 7
Therefore, the equation of the line that is perpendicular to y = 2x + 6 and passes through (4, 5) is y = (-1/2)x + 7.
(4, 5), m = -1/2.
Y = mx + b.
5 = (-1/2)4 + b,
b = 7.
Eq: Y = (-1/2)x + 7.