Find k so that the line through (4, 1) and (k, 3) is
a. parallel to 8x+9y=18
y= -8/9x + 8/9
b. perpendicular to 3x+4y=8
y= -3/4x +2
I have solved both equation and found the slope, but I am not sure were to go from there. When I tried it the first time I got -17/2 for K. Is this right or did I mess up? The "K" is throwing me so please let me know what I did wrong. Thank you!
a.
9 y = -8x + 18
y = -(8/9) x + 2
slope = -8/9
so
(3-1)/(k-4) = -8/9
2(9) = -8(k-4)
18 = -8k+32
8 k = 14
k = 7/4
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check
-8/9 = ? (3-1)/(7/4 -4)
= 2/(7-16)/4 = 8/-9 sure enough
now in b we get the slope first
yes, -3/4
so the slope we want is
-1/m = 4/3
now
4/3 = (3-1)/(k-4)
4 k- 16 = 6
4 k = 22
k = 11/2
To find the value of k that satisfies the given conditions, we'll use the fact that parallel lines have the same slope and perpendicular lines have slopes that are negative reciprocals.
To begin, let's find the slope of the line passing through (4, 1) and (k, 3) using the formula:
slope = (y2 - y1) / (x2 - x1)
Substituting the coordinates of the points into the equation, we get:
slope = (3 - 1) / (k - 4)
slope = 2 / (k - 4)
a. To find k so that the line is parallel to 8x + 9y = 18, we need to compare the slope of the line determined by (4, 1) and (k, 3) with the slope of the given line. The given line can also be written in slope-intercept form as y = -8/9x + 8/9. Comparing the two slopes:
2 / (k - 4) = -8/9
To solve for k, we can cross multiply:
9 * 2 = -8 * (k - 4)
18 = -8k + 32
-8k = 14
k = -14/8
k = -7/4
Therefore, the line through (4, 1) and (k, 3) is parallel to 8x + 9y = 18 when k = -7/4.
b. To find k so that the line is perpendicular to 3x + 4y = 8, we'll again compare slopes. The given equation can be rewritten as y = -3/4x + 2. Comparing the slopes:
2 / (k - 4) = -3/4
To solve for k:
4 * 2 = -3 * (k - 4)
8 = -3k + 12
-3k = 4
k = -4/3
Therefore, the line through (4, 1) and (k, 3) is perpendicular to 3x + 4y = 8 when k = -4/3.
In your attempt, you got k = -17/2, which is incorrect for both cases. Remember to be careful with your calculations and double-check your work.
So, the correct values of k are k = -7/4 for the parallel case and k = -4/3 for the perpendicular case.