Two lines L1 and L2, are perpendicular. The equation of L1 is 3x-y=2. L2 passes through (-5, -1)
Do you need to find L2?
If two lines are perpendicular, the slope m of one is the negative reciprocal of the slope of the other,
Slope m1 * m2 = -1
or, slope m2 = -1/m1
Your need to find slope m1 to write the equation of the perpendicular line.
L1 = 3x - y = 2
To find slope m, put the equation in slope-intercept form
y = mx + b, where m = slope and b = y-intercept
3x - y = 2
y = 3x - 2
So, slope m1 = 3
Since, m2 = -1/m1 and m1 = 3,
m2 = -1/3 = slope of L2 (perpendicular line)
L2 through point (-5, -1)
Form of the equation is,
y = mx + b
You found the slope m2 = -1/3
y = -1/3 x + b
To find b, use point (-5, -1) and substitute x and y point values in the equation and solve for b
y = -1/3 x + b
y = -5, x = -1
-1 = -1/3 (-5) + b
-1 = 5/3 + b
b = -1 + -5/3
b = -3/3 + -5/3
b = -8/3
So, L2 is
y = -1/3 x + -8/3
y = -1/3 x - 8/3
To find the equation of line L2, which is perpendicular to line L1, we need to use two pieces of information:
1. The slope of line L1.
2. The given point that line L2 passes through.
First, we need to find the slope of line L1. The equation of a line can be rearranged into the slope-intercept form, y = mx + b, where m is the slope of the line. The given equation of line L1 is 3x - y = 2. Let's rearrange it to this form:
- y = -3x + 2
y = 3x - 2
From this form, we can identify that the slope of line L1 is 3.
Since line L2 is perpendicular to line L1, the slopes of the two lines are negative reciprocals of each other. To find the slope of line L2, we take the negative reciprocal of 3, which is -1/3.
Now, we have the slope (-1/3) and the given point (-5, -1). We can use the point-slope form to get the equation of line L2. The point-slope form of a line is:
y - y1 = m(x - x1)
Substituting the values we have:
y - (-1) = (-1/3)(x - (-5))
Simplifying further:
y + 1 = (-1/3)(x + 5)
Distributing the -1/3:
y + 1 = (-1/3)x - 5/3
Finally, let's rearrange the equation into the standard form:
(-1/3)x - y = 1 + 5/3
Multiplying by 3 to eliminate the fraction:
- x - 3y = 3 + 5
Simplifying:
- x - 3y = 8
Therefore, the equation of line L2 is -x - 3y = 8.