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One Equation with several unknown quantities.

the difference of the squares is then just equal to the square of a.

The enunciation is corrected for this case by stating it as in example 48.

135. Corollary. It follows from example 7 of the preceding section that a fraction or ratio, which is greater than unity, is increased by diminishing both its terms by the same quantity; and a fraction or ratio, which is less than unity, is diminished by diminishing both its terms by the same quantity; but the reverse is the case, when the terms are increased instead of being diminished.

SECTION IV.

Equations of the First Degree containing two or more unknown quantities.

136. In the solution of complicated problems involving several equations, it is often found convenient to use the same letter to denote similar quantities, accents or numbers being placed to its right or left, above or below, so as to distinguish its different val

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may all be used to denote different quantities, though they generally are supposed to imply some similarity between the

Indeterminate Equations referred to the theory of Numbers.

quantities which they represent. Care must be taken not to confound the accents and the numbers in parentheses at the right with exponents.

137. Problem. To solve an equation with several unknown quantitics.

Solution. Solve the given equation precisely as if all its unknown quantities were known, except any one of them which may be chosen at pleasure; and in the value of this unknown quantity, which is thus obtained in terms of the other unknown quantities, any values whatever may be substituted for the other unknown quantities, and the corresponding value of the chosen unknown quantity is thus obtained.

138. Corollary. An equation which contains several unknown quantities is not, therefore, sufficient to determine their values, and is called indeterminate.

139. Scholium. The roots of an indeterminate equation are sometimes subject to conditions which cannot be expressed by equations, and which limit their values; such, for instance, as that they are to be whole numbers. But their investigation depends, in such cases, upon the particular properties of different numbers, and belongs, therefore, to the Theory of Numbers.

140. Theorem. Every equation of the first degree can be reduced to the form

Ax+By+C z + &c. + M = 0;

in which A, B, C, &c. and M are known quantities,

Solution of any Equation of the First Degree.

either positive or negative, and x, y, z, &c. are the unknown quantities.

Proof. When an equation of the first degree is reduced, as in art. 118, the aggregate of all its known terms may be denoted by M. Each of the other terms must have one of the unknown quantities as a factor; and, by art. 106, only one of them, and that one taken but once as a factor. Taking out, then, each unknown quantity as a factor from the terms in which it occurs, and representing its multiplier by some letter, as A, B, C, &c., the corresponding unknown quantities being represented by x, y, z, &c., the equation becomes

Ax+By+C z + &c. + M = 0.

141. Problem. To solve any equation of the first degree.

Solution. Having reduced the equation to the form
Ax+By+Cz + &c. + M = 0,

find, as in art. 137, the value of either of the unknown

quantities, as z, for instance, which is, by art. 121,

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and any quantities at pleasure may be substituted for y, z, &c.

142. Problem. To solve several equations with several unknown quantitics.

First Method of Solution called that of Elimination by Substitution. Find the value of either of the unknown quantities in one of the equations in which it occurs, and substitute its value thus found, which is generally in terms of the other unknown quantities, in all the other equations in which it occurs.

Solution of Equations. Elimination by Substitution.

The new equations thus formed, together with those in which this unknown quantity does not occur, are one less in number than the given equations, and contain one unknown quantity less, and may, by a succession of similar eliminations be still farther reduced in number and in the number of their unknown quantities, until only one equation is finally obtained; and the solution of all the given equations is thus reduced to that of one equation.

143. Corollary. When there are just as many equations as unknown quantities, the final equation of the preceding solution will, in general, contain but one unknown quantity, the value of which may be thence obtained; and this value, being substituted in the values of the other quantities, will lead to the determination of the values of all the unknown quantities.

144. Corollary. When the number of unknown quantities is more than that of the given equations, the final equation will contain several unknown quantities, and will therefore be indeterminate; so that a problem is indeterminate, which gives fewer equations than unknown quantities.

145. Corollary. When the number of unknown quantities is less than that of the given equations, only as many of the given equations are required to determine the values of the unknown quantities as there are unknown quantities; and the problem is therefore impossible, when the values of the un

Case in which the roots of two equations are Zero.

known quantities determined from the requisite equations do not satisfy the remaining equations.

146. Problem. To solve two equations of the first degree with two unknown quantities.

Solution. Suppose, as in art. 140, the given equations to be reduced to the forms

Ax+By+ M = 0,

A' x + By + M' = 0 :

in which x and y are the unknown quantities.

The value of x, obtained from the first of these equations, is

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The value of y is found from this equation, by art. 121,

to be

A' M-AM'

AB-A' B'

;

which, substituted in the above value of z, gives

BM-B' M

A B' - A' Β ́

147. Corollary. The value of x, obtained by the preceding solution would be zero, if its numerator were zero, that is, if

BM' = B' M.

But, in this case, if the first of the given equations is multiplied by B', and the second by B, these products become, by transposition and substitution,

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