If the points (4,2) and )-1, k) are on a line that is perpendicular to the line y=2x+1, what is the value of k?
To find the value of k, we need to first determine the slope of the line given by y = 2x + 1.
In this equation, the coefficient of x, which is 2, represents the slope of the line. Therefore, the slope of the line y = 2x + 1 is 2.
A line perpendicular to this line will have a slope that is the negative reciprocal of 2. The negative reciprocal is found by inverting the fraction and changing its sign. In this case, the negative reciprocal of 2 is -1/2.
So, we now have the slope of the perpendicular line. Let's use the point-slope form of a line, which is y - y1 = m(x - x1), to find the equation of the perpendicular line passing through the point (4, 2).
Substituting the values into the point-slope form, we have:
y - 2 = -1/2(x - 4)
Simplifying the equation, we get:
y - 2 = -1/2x + 2
To isolate y, let's add 2 to both sides:
y = -1/2x + 4
Now that we have the equation of the perpendicular line, we can substitute the x-coordinate of the point (-1, k) into the equation and solve for k.
Substituting x = -1 and y = k, we get:
k = -1/2(-1) + 4
Simplifying the equation, we have:
k = 1/2 + 4
Adding 1/2 and 4, we find:
k = 4.5
Therefore, the value of k is 4.5.
the slope of the given line is 2 ... y = m x + b , m is the slope
perpendicular lines have negative-reciprocal slopes
... so the new line slope is -1/2
(2 - k) / (4 - -1) = -1/2 ... solve for k