Unformatted text preview: ly, we
see that the lines 6x + 3y = 9 and 4x + 2y = 12 are distinct and parallel, and as such do not
intersect.
6. We can begin to solve our last system by adding the ﬁrst two equations
x−y = 0
+ (x + y = 2)
2x = 2
which gives x = 1. Substituting this into the ﬁrst equation gives 1 − y = 0 so that y = 1.
We seem to have determined a solution to our system, (1, 1). While this checks in the
ﬁrst two equations, when we substitute x = 1 and y = 1 into the third equation, we get
−2(1)+(1) = −2 which simpliﬁes to the contradiction −1 = −2. Graphing the lines x − y = 0,
x + y = 2, and −2x + y = −2, we see that the ﬁrst two lines do, in fact, intersect at (1, 1),
however, all three lines never intersect at the same point simultaneously, which is what is
required if a solution to the system is to be found.
y y
6
5
4
3
2
1
−1
−2
−3 1 1 2 6x + 3y = 9
4x + 2y = 12 x x
−1 y−x=0
y+x=2
−2x + y = −2 A few remarks about Example 8.1.1 are in order. It is clear that some systems of equations have
solutions, and some do no...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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