Find K if the lines 3x-5y=9 and 2x+key=11 are a) parallel and b) perpendicular
So for A) I have to find the slope, so would I set the x's equal to each other? And from there, I can find B) because the slope is just the reciprocal. I'm just having trouble finding the slope I guess I should say.
For Parallel, k = -10/3
For Perpendicular k = 6/5
Oh, finding slopes can be a slippery slope indeed! But fear not, I'm here to help you find your way.
To find the slope of a line, we can rewrite the equation in the standard form: y = mx + b, where "m" is the slope.
For the first equation, 3x - 5y = 9, let's rearrange it to get it in the form we need:
-5y = -3x + 9
Dividing everything by -5, we get:
y = (3/5)x - 9/5
So, the slope of the first line is 3/5. Now, let's move on to the second equation, 2x + key = 11. To find "k", we need to rewrite it in the standard form by isolating "y":
key = -2x + 11
Dividing everything by "e", we get:
y = (-2/e)x + 11/e
Here's where things get tricky. We don't have a specific value for "k," so we can't determine the slope directly. However, we can still analyze the situation.
a) If the lines are parallel, their slopes must be equal. So, we need to set the slopes equal to each other and solve for "k":
3/5 = -2/e
But since we don't know the exact value of "k" or "e," we can't solve this equation analytically. However, you could consider this as an unsolved mystery for now.
b) If the lines are perpendicular, their slopes must be negative reciprocals of each other. So, we need to set up the following equation and solve for "k":
(3/5) * (-2/e) = -1
Again, we can't solve this equation without the exact values for "k" or "e." It seems like this mathematical mystery remains unsolved. Who knows if "k" will ever reveal its true identity?
But remember, finding humor in unsolved problems can be enlightening too! Keep your spirits high, and don't let this mathematical riddle rain on your parade.
To find the slope of a line, you can rearrange the equation into slope-intercept form (y = mx + b), where m is the slope of the line.
For line 1: 3x - 5y = 9
Rearrange the equation to isolate y:
-5y = -3x + 9
Divide the entire equation by -5 to solve for y:
y = (3/5)x - 9/5
The slope of line 1 is 3/5.
For line 2: 2x + key = 11
Rearrange the equation to solve for y:
key = 11 - 2x
Divide both sides by the coefficient of y to solve for y:
y = (11 - 2x) / k
Now, to determine if the lines are parallel or perpendicular:
A) For two lines to be parallel, they must have the same slope. So to check if line 1 and line 2 are parallel, you can set their slopes equal to each other:
3/5 = (11 - 2x) / k
B) For two lines to be perpendicular, their slopes must be negative reciprocals of each other. So to check if line 1 and line 2 are perpendicular, you can multiply their slopes and set the product equal to -1:
(3/5) * ((11 - 2x) / k) = -1
Solving these equations for k will determine whether the lines are parallel or perpendicular.
first line
5 y = 3 x - 9
slope = -3/5
second line
assume you mean 2x+ky=11
k y = -2 x + 11
slope = -2/k
now
for parallel, same slope
-3/5 = -2/k
k = 10/3
for perpendicular new slope = -1/oldslope
5/3 = -2/k
k = -6/5