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EQUATIONS. The investigation of the properties of equations and the methods of the solution constitute one of the most important portions of the science of Algebra. An equation has been defined to be the equality of two algebraical expressions, denoting some relation between the known and unknown quantity or quantities involved in the equation. The expressiops “known quantities” and “unknown quantities' are terms employed in equations; the former denoting those which are supposed to be known or assignable, and the latter those which are to be determined. The unknown quantities are always dependent for their values on the known quantities involved in the equations, and the solution of equations is simply confined to determine “the unknown quantities” in terms of “the known quantities."
By means of the notations assumed to designate known and unknown quantities, it is always possible to distinguish what is given from what is required, or in other words the data and quæsita of a problem.
By the solution of an equation or a set of equations, is meant the process whereby the value or values of the unknown quantity or quantities can be determined in terms of the known quantities.
An equation may involve one unknown quantity, or more than one; and all equations may be separated into two general classes, determinate equations and indeterminate equations, the former class including equations which admit of a limited number of values of the unknown quantities, and the latter an unlimited number.
Equations both determinate and indeterminate may not admit of any intelligible solution. This arises from some inconsistent relations either in the data or in the quæsita, or in both of them, and will always become manifest in the results of the solution.
Every single equation expresses a condition-some specific relation between the known and unknown quantities, and the number of given independent equations may be equal to, greater than, or less than the number of the unknown quantities involved.
When the number of given independent equations is equal to the number of the unknown quantities involved in the equations, the equations are determinate, or limited to certain single values of the unknown quantities. If there be two independent equations, each involving two unknown quantities, in this case two unknown quantities are to be determined subject to two independent conditions. Between the two given equations one of the unknown quantities can be eliminated, and a third equation is found involving only one unknown. quantity, which can be determined in terms of the known quantities. The other un. known quantity is also determined by substituting in either of the given equations the value of that already determined. Hence it appears that two independent equations involving two unknown quantities are sufficient to determine the values of the two unknown quantities.
In the same manner, if there be three independent equations involving three unknown quantities; the first unknown quantity can be eliminated between the first and second equations, also between the first and third or second and third equations; and thus are derived two equations, each involving two unknown quantities, which case has been considered. Hence it is manifest that three independent equations involving three unknown quantities, are sufficient to determine the three unknown quantities.
When the number of equations is greater than the number of unknown quantities; if some of the equations can be derived from others so that the number of independent equations is really equal to the number of unknown quantities, the equations admit of a definite solution. If, however, all the equations are independent, they may express incompatible conditions, and every set of the equations, equal to the number of unknown quantities involved, will give different sets of solutions.
If the number of independent equations be less than the number of unknown quantities, the given conditions are insufficient to determine the unknown quantities, and they will therefore be indeterminate.
Equations may also, be divided into different orders. An equation of the first, second, i third, fourth, &c., order, is said to be of one, two, three, four, &c., dimensions, according as the highest power of the unknown quantity or quantities involved is the first, second, third, fourth, &c., power.
Equations of the first order, which involve when reduced only the first power of the unknown quantity, are named simple equations.
Equations which involve the second power of the unknown quantity, with or without the first power, are named quadratic equations.
Equations which involve the third power, with or without inferior powers of the unknown quantity or quantities, are named cubic equations.
Equations which involve the fourth power of the unknown quantity or quantities, with or without inferior powers, are named biquadratic equations.
And equations which involve higher powers of the unknown quantity, are named according to the index of the highest power of the unknown quantity involved in the equations.
Any process, whereby any quantity common to two or more equations, can be made to disappear in the resulting equation or equations, is called “the elimination of that quantity."
The elimination of quantities from equations is an important process, essential in the solution of all equations which involve more than one unknown quantity.
When there are two independent equations with two unknown quantities, x and y; if one of them, x, be eliminated, the resulting equation will involve only y, the other unknown quantity.
When there are three independent equations with three unknown quantities, x, y, z; any one of them, x, can be eliminated between tho first and second, the first and third, and between the second and third equations; and each of the three resulting equations involves y and %. As the values of x, y, z in the given equations are determinate, it follows that the values of y and z are the same in each of these last three equations, and any one of these equations is derivable from the other two, and consequently that there are only two independent equations resulting from the elimination.
Also from the two equations involving y and 2, if y, one of them, be eliminated, the final equation will involve , one unknown quantity. And this final equation may involve the first, second, third, &c., powers, of , depending on the powers of x, y, -, the unknown quantities in the original equations.
If there be four independent equations, involving four unknown quantities, by the successive elimination of one of them, a final equation will at length be obtained involving only one unknown quantity.
When two equations involve two unknown quantities, x and y; if when y is changed into x and x into y in the two equations, they retain the same form, they are called simultaneous equations, from the mutual interchange of the two unknown quantities.
Also when there are three equations and three unknown quantities, x, y, z; if after the mutual interchange of these quantities in the three equations, they retain the same form, they also are simultaneous equations.
The value of the unknown quantity in an equation is named the root of the equation. To solve an equation is to determine its root or roots, there be more than one. It is always possible to ascertain whether the root found by any process is the true value of the unknown quantity, as this quantity when substituted in the given equation for the unknown quantity always reduces the equation to an identity, or to the form () = 0, if all the terms are brought to one side of the equation. And this value of the unknown quantity is said " to satisfy the equation,” as 3 satisfies the equation 5x+7 = 22; for when 3 is substituted for x, both sides of the equation become identical in value, thus 15+7 = 22 or 22 = 22. And when 3 is substituted for x in the equation x–5x+6= 0, it becomes 9-15 +6 = 0 or 0=0.
And it is important to remark that this is the criterion by which it is known that the true root of an equation has been determined. In problems proposed for solution by means of equations the first step is to consider the nature of the question, and to discriminate accurately what is given, and what is required to be found, and to reject every condition
or circumstance which has no necessary connection or relation to the result required.
The next step is to assume letters to denote the data and quæsita of the problem, taking care to avoid ambiguity or contradiction in the assumptions.
After this designation of the known and unknown quantities, the problem must, with the aid of symbols of operation, be translated into algebraical language; that is, the conditions of the problem must be denoted by equations, making no distinction between the known and unknown quantities; and when this is effected it will then appear whether the problem is determinate or indeterminate.
Having formed the equations, the last process consists in their solution. It will be found that some methods are more operose and tedious than others. The most direct and simple methods of solution are to be preferred when such are possible; but these are not always subject to any general rule, but rather depend on the skill which is acquired by experience.
In the solution of equations involving more than one unknown quantity, it will sometimes be found that they are so involved that they cannot easily be separated by any of the ordinary methods. In such cases some artifice is generally possible, by means of which the equations may become simplified and their solution rendered practicable. It is impossible to lay down any rules for the employment of artifices in the solution of equations. The manner in which the unknown quantities are involved only can suggest what artifices are possible for rendering the equations more easy of solution. Instead of assuming single letters to denote unknown quantities, it will be found convenient in some cases to :assume for them the sum and difference of the two unknown quantities, in others the product and the quotient, &c.
All homogeneous equations of two unknown quantities may be simplified by assuming for one of them some unknown multiple of the other, and making the substitution in the two equations. By this means both of the unknown quantities can be eliminated, and the resulting equation will involve only the quantity which denotes the unknown multiple.
The same method can be extended to equations involving three or more than three unknown quantities.
It has been pointed out that equations proposed for solution may be deducible from, or contradictory to, one another. In the former case, in the attempted solution, some quantity may be found equal to itself, and in the latter some absurdity will follow, as a greater quantity equal to a less. And this dependency or inconsistency of equations is not always obvious until the identity or absıirdity appears in the final result of the operation.