Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form.
a. line 3x+y=5, point (-1,2)
b. line y=3x+1, point (6,1)
do like the other one, but use the perpendicular slope.
3x+y=5 has slope -3
so, the perpendicular line has slope 1/3
The new line is thus
y-2 = 1/3 (x+1)
now rearrange to slope-intercept form
do the other one in like wise.
To find an equation of a line that is perpendicular to the given line and passes through the given point, we need to follow a few steps.
Step 1: Find the slope of the given line.
The slope-intercept form of a line is y = mx + b, where m is the slope of the line.
For the given line, we need to rearrange the equation into slope-intercept form, y = mx + b.
a. line 3x + y = 5:
Subtract 3x from both sides: y = -3x + 5
The slope m is -3.
b. line y = 3x + 1:
The slope m is 3.
Step 2: Find the slope of the line perpendicular to the given line.
The slope of a line perpendicular to another line is the negative reciprocal of the slope of the given line.
To get the negative reciprocal of a number, flip the fraction and change the sign.
a. For the line perpendicular to a m = -3, the slope will be m' = 1/3.
b. For the line perpendicular to a m = 3, the slope will be m' = -1/3.
Step 3: Use the point-slope form to find the equation of the perpendicular line.
The point-slope form of a line is y - y1 = m'(x - x1), where (x1, y1) is the given point and m' is the slope perpendicular to the given line.
a. For a point (-1, 2) and a slope m' = 1/3:
Plug in the values: y - 2 = 1/3(x - (-1))
Simplify and rearrange: y - 2 = 1/3(x + 1)
Convert to slope-intercept form: y = 1/3x + 1/3 + 2
y = 1/3x + 7/3
b. For a point (6, 1) and a slope m' = -1/3:
Plug in the values: y - 1 = -1/3(x - 6)
Simplify and rearrange: y - 1 = -1/3(x - 6)
Convert to slope-intercept form: y - 1 = -1/3x + 2
y = -1/3x + 3
So, the equations of the lines perpendicular to the given lines and containing the given points are:
a. y = 1/3x + 7/3
b. y = -1/3x + 3