Write an equation of a line perpendicular to y=2x+3 and passes through (3,4).
Slope of the given line is 2 , therefore the slope of the perpendicular line is -(1/2)
Use your point slope equation y -y =m(x - x)
Y - 4 = -(1/2)(x - 3)
2(y - 4) = -1(x - 3). You are looking a proportion and cross multiplying
2y - 8 = -1x + 3
2y. =. -1x + 11
y. = -(1/2)x + (11/2)
perpendicular to y = 2x + 3 implies the new equation must be
y = (-1/2)x + b
sub in the point (3,4)
4 = (-1/2)(3) + b
b = 4 + 3/2 = 11/2
y = (-1/2)x + 11/2
To find an equation of a line perpendicular to a given line, we need to follow a few steps:
1. Determine the slope of the given line. The equation y = 2x + 3 is in slope-intercept form (y = mx + b), where the coefficient of x, which is 2 in this case, represents the slope of the line. So, the given line has a slope of 2.
2. Find the negative reciprocal of the slope of the given line. The negative reciprocal of 2 is -1/2. The negative reciprocal is obtained by flipping the fraction and changing its sign.
3. Use the slope (-1/2) and the given point (3, 4) to determine the equation of the line using the point-slope form. The point-slope form of a linear equation is y - y₁ = m(x - x₁), where m is the slope, and (x₁, y₁) is the given point. Substituting the values, we have: y - 4 = (-1/2)(x - 3).
4. Simplify the equation. Distribute the (-1/2) to (x - 3) to get: y - 4 = (-1/2)x + 3/2.
5. To obtain the equation in slope-intercept form, isolate y by adding 4 to both sides of the equation: y = (-1/2)x + 3/2 + 4.
6. Simplify further: y = (-1/2)x + 11/2.
So, the equation of the line perpendicular to y = 2x + 3 and passing through the point (3, 4) is y = (-1/2)x + 11/2.