These values constitute a solution of equations (1) and (3), but not of (1) and (2). For they form a solution of the L. C. D. (equated to 0) of the fractions in (2); that is, of

We conclude, therefore, that equations (1) and (2) do not have any solution.

It can, in fact, be shown that they are inconsistent. For (1) is equivalent to

or to

3x-3+4y 8 = 0,

3(x − 1) + 4 (y — 2) = 0.

Dividing both members of the last equation by (x − 1) (y − 2), we obtain

Equation (5) is evidently inconsistent with (2).

It should be noticed that in clearing (2) of fractions no unnecessary factor was used. The explanation of the apparent contradiction of the principle proved in Ch. X., Art. 3, is that this principle holds only when the fractional equation contains but one unknown number.

EXERCISES VII.

Solve the following systems of equations by the methods given in this

chapter:

=

= 4,

3

2

36.

2x

+16 y 10 x 881.

((a - b)x + (b −c)y + (c− a)z = 2(a2+b2+ c2-ab-ac-bc),

-

-

(a - b)y + (b − c) z + (c − a) x = ab + ac + bc — a2 — b2 — c2, x + y + z = 0.

CHAPTER XIV.

PROBLEMS.

1. As was stated in Ch. V., Art. 2 (ii.), every problem which can be solved must contain as many conditions, expressed or implied, as there are required numbers. In solving a problem by means of one equation in one unknown number, one of the required numbers was usually taken as the unknown number of the equation. All but one of the conditions of the problem were used to express the other required numbers in terms of the one selected as the unknown number. The remaining condition furnished the equation of the problem.

But a problem which contains more than one condition can be solved by means of a system of equations in which the unknown numbers are usually the required numbers of the problem. Each condition then furnishes an equation. The solution of the system of equations thus obtained gives the solution of the problem, if the conditions of the latter be consistent.

2. We will first solve by means of a system of two equations one of the problems which was solved in Ch. V. by means of one equation in one unknown number.

Pr. 1. (Pr. 4, Ch. V.) At an election at which 943 votes were cast, A and B were candidates. A received a majority of 65 votes. How many votes were cast for each candidate? stand for the number of votes cast for A,

Let

and

y for the number of votes cast for B.

Then, by the first condition,

x+y=943;

and by the second condition,

x - y = 65.

(1)

(2)