If equation 1 is perpendicular to equation 2, rearrange and find the value of m.
Equation 1: 3=9x−2y
Equation -8=mx+3y
What they want you to do:
#1 ..... 3=9x−2y
2y = 9x - 3
y = (9/2)x - 3 ---> slope = 9/2
#2 ..... -8=mx+3y
-mx - 8 = 3y
(-m/3)x - 8/2 = y ----> slope = -m/3
but they are perpendicular, so
-m/3 = -2/9
solve for m
general slope-intercept form is ... y = m x + b ... m is slope , b is y-intercept
#1 ... adding 2y and subtracting 3 ... 2y = 9x - 3
... dividing by 2 ... y = 9/2 x - 3/2
#2 ... subtracting mx and dividing by 3 ... y = -m/3 x - 8/3
perpendicular means negative-reciprocal slopes
... 3/m = 9/2 ... 9 m = 6
To determine if equation 1 is perpendicular to equation 2, we need to examine the relationship between their slopes.
The slope-intercept form of a linear equation is y = mx + c, where m represents the slope of the line.
Given equation 1: 3 = 9x - 2y
Rearranging equation 1 to the slope-intercept form gives us:
-2y = 9x - 3
y = (9/2)x - 3/2
Comparing equation 1 to the standard slope-intercept form, we can determine that the slope (m1) of equation 1 is 9/2.
Given equation 2: -8 = mx + 3y
We need to rearrange equation 2 to the slope-intercept form:
3y = -mx - 8
y = - (m/3)x - 8/3
Comparing equation 2 to the standard slope-intercept form, we can determine that the slope (m2) of equation 2 is -m/3.
For equation 1 and equation 2 to be perpendicular, their slopes must be negative reciprocals of each other.
Therefore, we can set up the equation:
m1 * m2 = -1
Substituting the values of m1 and m2, we get:
(9/2) * (-m/3) = -1
Now we can solve for m:
(9/2) * (-m/3) = -1
(9/2) * (-1/3) * m = -1
(-9/6) * m = -1
Multiplying both sides by -6/9 to solve for m, we get:
m = -6/9
Simplifying further by dividing both numerator and denominator by 3:
m = -2/3
So the value of m is -2/3.
To determine the value of slope `m` such that Equation 1 is perpendicular to Equation 2, we need to use the concept that perpendicular lines have slopes that are negative reciprocals of each other.
First, let's find the slope of Equation 1 by rearranging it in slope-intercept form (y = mx + b), where `m` represents the slope:
3 = 9x - 2y
Rearranging the equation, we get:
2y = 9x - 3
Dividing both sides by 2, we have:
y = (9/2)x - 3/2
Comparing the equation with the slope-intercept form (y = mx + b), we find that the slope `m` of Equation 1 is 9/2.
Since the slopes of perpendicular lines are negative reciprocals, the slope of Equation 2 will be the negative reciprocal of 9/2. Let's denote it as `m2`.
To find `m2`, rearrange Equation 2:
-8 = mx + 3y
Rearranging the equation, we have:
3y = -mx - 8
Dividing both sides by 3, we obtain:
y = (-m/3)x - (8/3)
Comparing the equation with the slope-intercept form, we find that the slope `m2` of Equation 2 is -m/3.
Since Equation 1 is perpendicular to Equation 2, the slopes must be negative reciprocals:
m * (-m/3) = -1
Simplifying the equation, we have:
-m^2 = -3
Dividing both sides by -1, we get:
m^2 = 3
Taking the square root of both sides, we find:
m = ±√3
Therefore, the value of `m` for which Equation 1 is perpendicular to Equation 2 is ±√3.