Expert: Paul Klarreich - 5/17/2007

Question
hi i am trying this question. Find the values of m,n E R for which the system of equations x+2y=1; 2x+my=n  (a)possesses a unique solution (b)is inconsistent (c) possesses infinitely many solutions. I know that for uniqueness these lines must intersect; for (b) the lines will not intersect(will be parallel i think) and for (c) the two lines will have similar equations (or will basically be the same line) HOWEVER how do i go about proving the above conditions given the equations? thank you so very much in advance, i am greatly appreciative.  

Answer
Questioner:   jon
Category:  Advanced Math
 
Question:  hi i am trying this question. Find the values of m,n E R for which the system of equations x+2y=1; 2x+my=n  (a)possesses a unique solution (b)is inconsistent (c) possesses infinitely many solutions.

I know that for uniqueness these lines must intersect;

>> In only one point.  (You have to say that.) Then their slopes must not be the same.

for (b) the lines will not intersect(will be parallel i think)

>> so they will have the same slope, but not be identical.

and for (c) the two lines will have similar equations (or will basically be the same line)

>> yes, that is correct.

HOWEVER how do i go about proving the above conditions given the equations? thank you so very much in advance, i am greatly appreciative.

.................................
Hi, Jon,

Here are your equations, denoted I and II.

x + 2y = 1     (I)
2x + my = n    (II)

In general, the line  ax + by = c has a slope equal to -a/b

Equation I has a slope equal to -1/2.  
Equation II has a slope equal to  -2/m

Set those equal (or unequal) to each other.  Equals first:

-1/2 = -2/m

1/2 = 2/m

Cross-multiply:  m = 4.

We are in business.  If  m = 4, the lines have the same slope and are EITHER parallel (no intersection) or identical (many intersections).  If  m /= 4,

the lines have different slopes and a unique intersection.

Going further:  Suppose m = 4.  The equations are:

x + 2y = 1     (I)
2x + 4y = n    (II)

Divide out the '2' in II:

x + 2y = 1     (I)
x + 2y = n/2   (II)

These equations will be identical if n/2 = 1, or n = 2.  So that completes things:

m /= 4:  Unique solution.
m = 4  and  n = 2:  Many solutions -- lines coincide.
m = 4  and  n /= 2: No solution -- lines are parallel.