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Which of the following best describes the lines y-3x=4x and 6-2y=8x

○perpendicular
○parallel
○skew
○intersecting

1 Answer

3 votes

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

----------------------------------------------------------------------------------------------------------

-y = -7x

+

y = -4x + 3

----------------------------------------------------------------------------------------------------------

(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.

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