Unformatted text preview: ou imagine three sheets of notebook paper each representing a portion
of these planes, you will start to see the complexities involved in how three such planes can intersect.
Below is a sketch of the three planes. It turns out that any two of these planes intersect in a line,11
so our intersection point is where all three of these lines meet. 10 You were asked to think about this in Exercise 13 in Section 1.1.
In fact, these lines are described by the parametric solutions to the systems formed by taking any two of these
equations by themselves.
11 456 Systems of Equations and Matrices Since the geometry for equations involving more than two variables is complicated, we will focus
our eﬀorts on the algebra. Returning to the system 1
1 x − 1y + 2z =
y − 2z = z = −1
we note the reason it was so easy to solve is that the third equation is solved for z , the second
equation involves only y and z , and since the coeﬃcient of y is 1, it makes it easy to solve for y
using our known value for z . Lastly, the coeﬃcient of x in the ﬁrst equation is 1 making it easy to
substitute the known values of y and z and...
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