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Unformatted text preview: nd more variables in this, and
Consider the system of equations 1
1 x − 1y + 2z =
y − 2z = z = −1 1
Clearly z = −1, and we substitute this into the second equation y − 2 (−1) = 4 to obtain y = 7 .
Finally, we substitute y = 7 and z = −1 into the ﬁrst equation to get x − 3 7 + 1 (−1) = 1,
so that x = 3 . The reader can verify that these values of x, y and z satisfy all three original
equations. It is tempting for us to write the solution to this system by extending the usual (x, y )
notation to (x, y, z ) and list our solution as 8 , 7 , −1 . The question quickly becomes what does
an ‘ordered triple’ like 8 , 2 , −1 represent? Just as ordered pairs are used to locate points on the
two-dimensional plane, ordered triples can be used to locate points in space.10 Moreover, just as
equations involving the variables x and y describe graphs of one-dimensional lines and curves in the
two-dimensional plane, equations involving variables x, y , and z describe objects called surfaces
in three-dimensional space. Each of the equations in the above system can be visualized as a plane
situated in three-space. Geometrically, the system is trying to ﬁnd the intersection, or common
point, of all three planes. If y...
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