Answer:
Intersecting (fourth answer choice)
Explanation:
- If the lines are perpendicular, parallel, or intersecting, they are not skew.
- Thus, we need to check if the lines can be classified as either perpendicular, parallel, or intersecting first.
- If the lines are classified as neither, then they are skew.
First, let's convert both lines to slope-intercept form, whose general equation is y = mx + b, where
- m is the slope,
- and b is the y-intercept.
Converting y - 3x = 4x to slope-intercept form:
(y - 3x = 4x) + 3x
y = 7x
Thus, the slope of this line is 7 and the y-intercept is 0.
Converting 6 - 2y = 8x to slope-intercept form:
(6 - 2y = 8x) - 6
(-2y = 8x - 6) / -2
y = -4x + 3
Thus, the slope of this line is -4 and the y-intercept is 3.
Checking if y = 7x and y = -4x + 3 are perpendicular lines:
- The slopes of perpendicular lines are negative reciprocals of each other.
We can show this in the following formula:
m2 = -1 / m1, where
- m1 is the slope of one line,
- and m2 is the slope of the other line.
Thus, we only have to plug in one of the slopes for m1. Let's do -4.
m2 = -1 / -4
m2 = 1/4
Thus, the slopes 7 and -4 are not negative reciprocals of each other so the two lines are not perpendicular.
Checking if y = 7x and y = -4x + 3 are parallel lines:
The slopes of parallel lines are equal to each other.
Because 7 and -4 are not equal, the two lines are not parallel.
Checking if the lines intersect:
- The intersection point of two lines have the same x and y coordinate.
- To determine if the two lines intersect, we treat them like a system of equations.
Method to solve the system: Elimination:
We can multiply the first equation by -1 and keep the second equation the same, which will allow us to:
- add the two equations,
- eliminate the ys,
- and solve for x:
-1 (y = 7x)
-y = -7x
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-y = -7x
+
y = -4x + 3
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(0 = -11x + 3) - 3
(-3 = -11x) / 11
3/11 = x
Now we can plug in 3/11 for x in y = 7x to find y:
y = 7(3/11)
y = 21/11
Thus, x = 3/11 and y = 21/11
We can check our answers by plugging in 3/11 for x 21/11 for y in both y = 7x and y = -4x + 3. If we get the same answer on both sides of the equation for both equations, the lines intersect:
Checking solutions (x = 3/11 and y = 21/11) for y = 7x:
21/11 = 7(3/11)
21/11 = 21/11
Checking solutions (x = 3/11 and y = 21/11) for y = -4x + 3:
21/11 = -4(3/11) + 3
21/11 = -12/11 + (3 * 11/11)
21/11 = -12/11 + 33/11
21/11 = 21/11
Thus, the lines y = 3x = 4x and 6 - 2y = 8x are intersecting lines (the first answer choice).
This also means that lines are not skew since lines had to be neither perpendicular nor parallel nor intersecting to be skew.