Expert: Sherman D. - 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
x + 2y = 1
2y = -x + 1
y = (-1/2)x + (1/2)

2x + my = n
my = -2x + n
y = (-2/m)x + (n/m)

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a.)

y = (-2/m)x + (n/m)
y = (-1/2)x + (1/2)

Slope = (-1/2)

for uniqueness they need to be perpendicular, Slope = 2

-2/m = 2
m = -2/2
m = -1

y = (-2/m)x + (n/m)
y = (-2/-1)x + (n/-1)
y = 2x - n

m = -1
n = Any Value

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b.)
y = (-2/m)x + (n/m)
y = (-1/2)x + (1/2)

For inconsistancy, then need to be parallel without the same "n/m" value.

(-2/m) = (-1/2)
-4 = -m
m = 4

y = (-2/m)x + (n/m)
y = (-2/4)x + (n/4)
y = (-1/2)x + (n/4)

(n/4) �‚ (1/2)
2n �‚ 4
n �‚ 2

m = 4
n < 2 or n > 2, or you can probably just say n �‚ 2

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c.)
y = (-2/m)x + (n/m)
y = (-1/2)x + (1/2)

for this to possess infinite solution, they have to be exactly the same.

(-2/m)x + (n/m) = (-1/2)x + (1/2)

multiply everything by 2m

-4x + 2n = -mx + m

m = 4
n = 2

on this one, you just look at them

-4x = -mx
divide both sides my x

-m = -4
m = 4

and

(n/m) = (1/2)

and since i already stated "m" = 4

n/4 = 1/2
2n = 4
n = 2