Write an equation in point-slope form for the line that is perpendicular to the given line and passes through the given point.
𝑦 − 3 = 4(𝑥 + 2) through the point (−2,6)
you want a slope of -1/4, so
y-6 = -1/4 (x+2)
Review perpendicular slopes.
Well, if we want to find a line that is perpendicular to the given line, we first need to determine the slope of the given line. The given line equation is 𝑦 − 3 = 4(𝑥 + 2), which can be rewritten as 𝑦 = 4𝑥 + 11. The slope of this line is 4.
To find the slope of a line perpendicular to this, we use the fact that the product of the slopes of two perpendicular lines is -1. So, the slope of the line we are looking for is -1/4 (the negative reciprocal of 4).
Now that we know the slope (-1/4), and we are given a point (-2,6) that the line passes through, we can use the point-slope form of a line equation:
𝑦 − 𝑦₁ = 𝑚(𝑥 − 𝑥₁),
where 𝑚 is the slope and (𝑥₁, 𝑦₁) are the coordinates of the given point.
Plugging in the values, we get:
𝑦 − 6 = (-1/4)(𝑥 − (-2)).
Simplifying this equation will give you the equation of the line in point-slope form that is perpendicular to the given line and passes through the given point.
To find the equation of a line perpendicular to another line, we need to determine the negative reciprocal of the slope of the given line.
The given line has a slope of 4. The negative reciprocal of 4 is -1/4.
We also have the point (-2, 6) through which the perpendicular line passes.
Now we can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope.
Plugging in the values:
y - 6 = (-1/4)(x - (-2))
Simplifying:
y - 6 = (-1/4)(x + 2)
This is the equation of the line in point-slope form that is perpendicular to the given line and passes through the point (-2, 6).
To find the equation of a line that is perpendicular to another line, we need to determine the slope of the given line and then find the negative reciprocal of that slope.
Given line: y - 3 = 4(x + 2)
First, we need to rewrite the equation in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
y - 3 = 4(x + 2)
y - 3 = 4x + 8
y = 4x + 8 + 3
y = 4x + 11
From this equation, we can see that the slope of the given line is 4.
To find the slope of the line perpendicular to this, we take the negative reciprocal of 4. The negative reciprocal of a number is obtained by flipping the fraction and changing the sign.
Negative reciprocal of 4: -1/4
Now that we have the slope (-1/4) and the given point (-2, 6), we can use the point-slope form to find the equation.
The point-slope form of a line is given by the equation: 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 (-2, 6) and the slope -1/4, we get:
y - 6 = -1/4(x - (-2))
y - 6 = -1/4(x + 2)
y - 6 = -1/4x - 1/2
This is the equation in point-slope form for the line that is perpendicular to the given line and passes through the point (-2, 6).