Find the value of k so that the line containing the points (k,−8) and (−3,−5) is parallel to the line containing the points (9,7) and (12,5).
(k,-8), (-3.-5).
(9,7), (12,5).
m1 = m2 = (5-7)/(12-9) = -2/3.
m1 = (-5-(-8))/(-3-k) = -2/3
(-5+8)/(-3-k) = -2/3
3/(-3-k) = -2/3
6+2k = 9
2k = 3
k = 3/2.
To find the value of k that makes the line containing the points (k, -8) and (-3, -5) parallel to the line containing the points (9, 7) and (12, 5), we need to determine if the slopes of these two lines are equal.
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)
Let's first find the slope of the line passing through the points (9, 7) and (12, 5):
slope1 = (5 - 7) / (12 - 9)
= -2 / 3
Now, let's find the slope of the line passing through the points (k, -8) and (-3, -5):
slope2 = (-5 - (-8)) / (-3 - k)
= 3 / (-3 - k)
= -3 / (k + 3)
Since we want the lines to be parallel, the slopes must be equal:
-3 / (k + 3) = -2 / 3
To solve this equation for k, we can cross-multiply:
-2(k + 3) = -3 * (-3)
-2k - 6 = 9
Now, let's solve this equation for k:
-2k = 9 + 6
-2k = 15
k = 15 / -2
k = -7.5
Therefore, the value of k that makes the line containing the points (k, -8) and (-3, -5) parallel to the line containing the points (9, 7) and (12, 5) is k = -7.5.
To determine the value of k that makes the line containing the points (k, -8) and (-3, -5) parallel to the line containing the points (9, 7) and (12, 5), we need to compare the slopes of the two lines.
The slope between two points (x₁, y₁) and (x₂, y₂) can be calculated using the formula:
slope = (y₂ - y₁) / (x₂ - x₁)
For the line containing (k, -8) and (-3, -5):
slope₁ = (-5 - (-8)) / (-3 - k) = 3 / (k + 3)
For the line containing (9, 7) and (12, 5):
slope₂ = (5 - 7) / (12 - 9) = -2 / 3
Since we want the two lines to be parallel, their slopes must be equal. Therefore, we can set the two slope formulas equal to each other and solve for k:
3 / (k + 3) = -2 / 3
To eliminate the fractions, we can cross multiply:
(3)(3) = (-2)(k + 3)
9 = -2k - 6
Next, we isolate k by moving the constant term to the right:
-2k = 9 + 6
-2k = 15
Finally, divide both sides of the equation by -2:
k = 15 / -2
k = -7.5
Therefore, the value of k that makes the line containing the points (k, -8) and (-3, -5) parallel to the line containing the points (9, 7) and (12, 5) is k = -7.5.