Mathematical Preliminaries for ECOS2001
Oleksii Birulin
2008
1
Linear equations
linear equation always has a form
ax
+
b
=
c;
where
a; b; c
are some numbers and
x
stands for the variable that we have to solve
for.
the way to solve a linear equation: °rst subtract
b
from both sides± we get
ax
=
c
°
b
now divide both sides by
a
and we get the answer
x
=
c
°
b
a
1.1
Examples
Suppose we have to solve
5
x
+ 8 = 10
take 8 from both sides we get
5
x
= 2
now divide by 5 and we get an answer
x
=
2
5
Another example
°
3
y
°
10 = 23
add 10 to both sides (or equivalently take 10 from both sides) we get
°
3
y
= 33
now divide by 3 we get an answer
y
=
°
11
1
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1.2
Additional exercises
Solve the equations
2
x
+ 5
=
10
4
x
°
5
=
16
1
3
x
+ 1
=
0
°
5
6
y
°
1
6
=
2
°
5
6
y
+
1
3
=
1
2
Systems of linear equations
We deal with systems of 2 equations only. In general the system of 2 linear equations
can be represented as
ax
+
by
+
c
=
d
ex
+
fy
+
g
=
h
where
a; b; c; d; e; f; g; h
are some numbers and
x; y
are the variables that we are
interested in.
note that to pin down 2 variables simultaneously we have to have 2 equations,
one is not enough. The easiest way to solve the system is use one of the equations
to °gure out
y
in terms of
x
and then substitute the expression that you get into
the remaining equation. What we obtain as a result of this substitution is a linear
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 Three '09
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 Elementary algebra, Additional Exercises, Oleksii Birulin

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