1 Introduction to systems of linear
equations
Basic terminology
Methods of solving linear systems
2 Linear equations
Recall that in two dimensions a line in the rectangular
xOy−coordinate system can be represented by an equation
of the form
ax + by = c (a and b not both 0)
and in three dimensions a plane in a rectangular xyz−coordinate
system can be represented by an equation of the form
ax + by + cz = d (a, b, c not all 0).
More generally, we define a linear equation in n variables
x1, x2, ..., xn to be one that can be expressed in the form
a1x1 + a2x2 + ... + anxn = b
where a1, a2, ..., an and b are constants, and the a’s are
not all zero.
Example 1 4x1 − 5x2 + 2 = x1
rearranged
→
3x1 − 5x2 = −2
or
x2 = 2
³√6 − x1
´
+ x3
rearranged
→
2x1 + x2 − x3 = 2√6
Example 2 Not linear equation
4x1 − 5x2 + 2 = x1x2 or 4x1 − 5√x2 + 2 = 0
sin x1 + x2 = 0
Observe that in a linear equation all the variables occur
only to the first power and do not appear, for example,
as arguments of trigonometric, logarithmic or exponential
function.
Definition 3 The variables x1, x2, ..., xn are called unknowns.
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