Imágenes de páginas
PDF
EPUB

CHAPTER VIII

THE GENERAL SOLUTION OF A SYSTEM OF THREE EQUATIONS IN THREE UNKNOWN QUANTITIES THE SOLUTION OF THREE 'OR MORE EQUATIONS IN AS MANY UNKNOWN

QUANTITIES

238. The first step in solving a system of two equations in two unknown quantities, is the elimination of one of the unknown quantities; this elimination results in an equation in one unknown quantity which can at once be solved and which, with one of the given equations, forms a system equivalent to the given system of equations.

The solution of a system of three equations in three unknown quantities is an extension of the principle stated above.

First combine any two, say the first and the second, equations of the given system and eliminate one of the unknown quantities; then combine say, the second and the third equations, to eliminate the same unknown quantity. This gives two equations in two unknown quantities, which can be solved in the usual way, and the third unknown quantity can be found by substituting the two values found in any one of the three given equations.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]

To eliminate y from (5) and (6), multiply (5) by 9 and (6) by 4 and

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

Hence the given system has the solution,

(15) x=7, y = 5, ≈ = 4,

z=

and this solution only. The system of equations (15) is equivalent

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

To eliminate x from (1) and (2), subtract (2) from (1),

[merged small][merged small][ocr errors][merged small][ocr errors]

This equation contains y and z; so also does (3).

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

To eliminate y from (1), solve (2) for y, i. e., y = z c and substitute z c for

or

y

in (1); thus,

[blocks in formation]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

RULE. Hence, to solve three equations in three unknown numbers, eliminate one of the unknown numbers from any two of the equations, and eliminate the same number from any one of these and the third equation. These two steps give two equations in two unknown numbers.

Solve these two equations by the usual method; then substitute their values in the simplest of the three given equations; the third unknown number will be found by solving the resulting equation.

239. To solve a system of four equations in four unknown quantities, solve one of the equations for one of the unknown numbers and substitute this value in each of the other three equations; we then have, instead of the given system, an equivalent system of four equations, three of which contain three unknown quantities. These three equations can be solved by the method already explained in the preceding paragraph.

[blocks in formation]

The substitution of this value of x in each of the given equations, gives the equivalent system II:

II

(6)

-2 z

u + 4y + 2 z = 23

(7) 3 (13+3 y−22) + 4 u − 2 z = 35 or 8 u +9 y — 10% = 31

2

(8) 4 (13+3 y −2 2) — 5 y + 3 u = 34 or 2 y + 6 u − 8 z = 16

[blocks in formation]

Equations (6), (7), and (8) may now be solved by the method

[blocks in formation]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

The work can often be shortened by introducing some simple device.

(19)

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

This example can be readily solved also by finding the sum of (1), (2), (3), and (4), then dividing by 3 and subtracting from the resulting equation each of the equations (1), (2), (3), and (4).

NUMBER OF SOLUTIONS OF A SYSTEM OF n LINEAR EQUATIONS

240. The examples which have been solved in the preceding section illustrate the following principles:

(1.) A system of n independent and compatible linear equations in n unknown numbers has one, and only one, determinate solution.

Deduce from the first equation the value of the first unknown number x, as though the others were known, and substitute this value for x in each of the n - 1 other equations; thus a system equivalent to the first is obtained, composed, first, of one equation in n unknown numbers, second, of n -1 equations in n- - 1 unknown numbers.

« AnteriorContinuar »