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 
