BTGModeratorVI wrote: ↑

Mon Jun 22, 2020 6:17 am

A given line L has an equation 3x+4y=5. Which of the following is the equation of line which does not intersect the above line?

(A) 4x + 3y = 5
(B) 3x + 4y = 10
(C) 3x + 5y = 5
(D) 3x + 5y = 3
(E) 3x – 4y = 5

Answer: B
Source: Veritas Prep

IMPORTANT: If a line intersects another line, then the coordinates of the point of intersection will satisfy the equations for BOTH lines.

Notice that line L tells us that 3x + 4y = 5. That is, the coordinates of ANY point on line L will satisfy the equation 3x + 4y = 5
The equation for answer choice B is 3x + 4y = 10. That is, the coordinates of ANY point on line B will satisfy the equation 3x + 4y = 10
As you can see, it's impossible for a set of coordinates to satisfy the equation 3x + 4y = 5 AND the equation 3x + 4y = 10
In one case, the sum of 3x and 4y is 5, and in the other case, the sum of 3x and 4y is 10. We can't satisfy both equations at the same time.
So, those two lines CANNOT share a common point.

Answer: [spoiler=]B[/spoiler]

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BTGModeratorVI wrote: ↑

Mon Jun 22, 2020 6:17 am

A given line L has an equation 3x+4y=5. Which of the following is the equation of line which does not intersect the above line?

(A) 4x + 3y = 5
(B) 3x + 4y = 10
(C) 3x + 5y = 5
(D) 3x + 5y = 3
(E) 3x – 4y = 5

Answer: B

Solution:

If two lines don’t intersect, they must be parallel. Let’s say one line has equation Ax + By = C and the other line has equation A’x + B’y = C’. Then these two lines are parallel if A’ = Ak and B’ = Bk for some nonzero constant k, but C’ ≠ Ck.

For example, if k = 1, then the two lines are parallel if A’ = A and B’ = B, but C’ ≠ C. We can see that the line with equation 3x + 4y = 10 (choice B) fits this condition, so it doesn’t intersect the line with equation 3x + 4y = 5 since they are parallel.

In other words, in this problem, if the two lines’ equations are identical except for the constant, they are parallel. Thus, B is the correct answer choice.

Alternate Solution:

If two lines don’t intersect, they must be parallel. If two lines are parallel, they have the same slope but different y-intercepts. In the slope-intercept form y = mx + b, the given line becomes y = (-3/4)x + 5/4. Let’s look for a line with the same slope but with a different y-intercept:

A) 4x + 3y = 5

In slope-intercept form, this line becomes y = (-4/3)x + 5/3. Since the slope of this line is not -3/4, this line is not parallel to 3x + 4y = 5.

B) 3x + 4y = 10

In the slope-intercept form, this line becomes y = -(3/4)x + 5/2. Since the slope of this line is -3/4 and the y-intercept is different from 5/4, this line is parallel to 3x + 4y = 5.

Answer: B