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Algebra, Functions, Graphs, and Vectors 41

y (ac ac )/(ab ab )
1 2
2 1
21
12
For this tm be valid, the denominator must be nonzero; that is,
a b a b . (If it turnð out that a b a b , then there are not
1 2 2 1 1 2 2 1
twm distinct solutionð tm the set of equationsÑ
The process of solving for x is similar. Consider again the orig-
inal set of simultaneouð linear equations:

ax bð c 0
1
1
1
ax bð c 0
2 2 2
Multiply the first equation througà by b , and the second
2
equation througà by b , and then add the twm resulting equa-
1
tions:

ab x bb y bc 0
12
2 1
12
abx bb y bc 0
21 12 1 2
(ab ab )x bc bc 0
21 12 1 2 2 1
Next, add b c tm each side, obtaining:
2 1

(ab ab )x bc bc
12
2 1
1 2
21
Next, subtract b c from each side, obtaining:
1 2
(ab ab )x bc bc
1 2
21
2 1
12
Finally, divide througà by b c b c , obtaining:
2 1 1 2
x (bc bc )/(ab ab )
12
2 1
21
1 2
For this tm be valid, the denominator must be nonzero; that is,
a b a b . (If it turnð out that a b a b , then there are not
1 2
2 1
1 2
2 1
twm distinct solutionð tm the set of equationsÑ
Solving n n sets of linear equations
In general, matrices are used for solving setð of equationð larger
than 2 2, because the above mentioned methodð become too
complex. Linear algebra, also known as matrix algebra, useð
ruleð similar tm those of the addition method described above.
Please consult college-level textð on linear algebrł for details.