CONTAINING MORE THAN TWO UNKNOWN QUANTITIES.

171. If we have three or more simultaneous equations, they may be reduced by successive eliminations, as follows:

INFERENCES.

1.—If we have n equations, and proceed as above to combine one of them with each of the others, eliminating the same letter by each combination, we shall have n-1 derived equations containing all of the unknown quantities except the one eliminated.

2. If then we combine one of these derived equations with each of the others, eliminating another letter, we shall have n-2 derived equations, containing all of the unknown quantities except the two eliminated quantities.

3. Since each succeeding group of derived equations consists of one less equation than the preceding group, it follows that if this process of successive elimination be continued, the (n-1)th group will consist of a single equation; and this will contain all of the unknown quantities except the n-1 eliminated quantities. Hence,

4.—If the number of original equations equals the number of unknown quantities, the final equation will contain but a single unknown quantity, the value of which may be found. By substituting this value in one of the equations of the preceding group, the value of a second unknown quantity may be determined; and so on.

5. But if the number of original equations is less than the number of unknown quantities, the final equation will contain more than one unknown quantity, and will be indeterminate, (162); and consequently, the given equations will be indeterminate.

6. In the solution of two or more simultaneous equations of the first degree, by successive eliminations, the value of each letter is determined finally by a simple equation containing only that letter. And since every such equation can have only one root, (155), it follows that any group of simultaneous equations can be satisfied by only one set of values of the unknown quantities.

172. From the foregoing inferences we derive the following

RULE. I. Combine one of the given equations with each of the others, eliminating the same unknown quantity by each combination; then combine one of the new equations with each of the others, eliminating a second unknown quantity, and thus continue till a final equation is obtained, containing but one unknown quantity.

II. Reduce this final equation, and find the value of the unknown quantity which it involves; substitute this value in an equation containing two unknown quantities, and thus find the value of a second; substitute these values in an equation containing three unknown quantities, and find the value of a third; and so on, till the values of all are found.

PRACTICAL SUGGESTIONS.

This rule may be modified in certain cases, as follows: :

1. Instead of combining the first equation with each of the others, we may pursue any order of combination, or adopt any one of the four methods of elimination, which seems best suited to the mutual relations of the coefficients. The following example will illustrate the precept just given:

2.-If two or more of the equations, taken together, contain less than all the unknown quantities, it is generally most convenient to employ these equations first, in the process of elimination. Thus,

3.- When the coefficients sustain to each other relations of equality or symmetry, it is often convenient to employ an auxiliary quantity, as in the two examples which follow:

Since in each equation one letter is wanting, let

1

EXAMPLES FOR PRACTICE.

Required the values of the unknown quantities in the following

NOTE.-In the last example, assume x+y+2=s, and add this equation to each of the given equations. Then determine s as in (3, 2).

26,

= 40,

y = 30, z= 24.

u = = 3,

x=

= 4,

y = 7,

z = 1.

5,

Ans.

y=

-8,

%=

4.