This value of y is not definitely determined. We may substitute in it any particular numerical value for x, and obtain a corresponding value for y. Thus, when x = 3, y=2; etc. In like manner the equation could have been solved for x in terms of y, and corresponding sets of values obtained. 1, y=4; when x = 2, y = 3; when x = Any set of corresponding values of x and y satisfies the given equation, and is therefore a solution. An equation which, like the above, has an indefinite number of solutions, is called an Indeterminate Equation. also has an unlimited number of solutions. Solving this equation for y, we have = У 1+x. Then, = 3; when x = when x = 1, y=2; when x = 2, y= 3, y=4; etc. Now, observe that equations (1) and (2) have the common solution, x2, y = 3. It seems evident, and we shall later prove, that these equations have only this solution in common. Equations (1) and (2) express different relations between the unknown numbers, and are called Independent Equations. Also, since they are satisfied by a common set of values of the unknown numbers, they are called Consistent Equations. = 3. The equations x + y 5 and 3x + 3y = 16 are not satis fied by any common set of values of x and y. For any set of values which reduces x + y to 5 must reduce 3x+3y, or 3 (x + y), to 15, and not to 16. These two equations express inconsistent relations between the unknown numbers, and are called Inconsistent Equations. 4. The three equations x+y=5(1), y − x = 1 (2), 2x+y=9(3), are not satisfied by any common set of values of x and y. For, by Art. 2, equations (1) and (2) are satisfied by the values x=2, y = 3. But equation (3) is evidently not satisfied by this set of values. The three equations express three independent relations between x and y. 5. A System of Simultaneous Equations is a group of equations which are to be satisfied by the same set, or sets, of values of the unknown numbers. A Solution of a system of simultaneous equations is a set of values of the unknown numbers which converts all of the equations into identities; that is, which satisfies all of the equations. The examples of Arts. 1-4 are illustrations of the following general principles, which will be proved later: A system of linear equations has a definite number of solutions, (i.) When the number of equations is the same as the number of unknown numbers. (ii.) When the equations are independent and consistent. EXERCISES I. 1. Are equivalent equations consistent? Are they independent? Which of the following systems have inconsistent equations? Which have equations not independent? Which have equations consistent and independent? § 2. EQUIVALENT SYSTEMS. 1. Two systems of equations are equivalent when every solution of either system is a solution of the other. E.g., the systems (I.) and (II.): are equivalent. For they are both satisfied by the solution, x = 2, y = 1, and, as we shall see later, by no other solution. 2. The solution of a system of equations depends also upon the following principles of the equivalence of systems: (i.) If any equation of a system be replaced by an equivalent equation, the resulting system will be equivalent to the given one. Thus, the systems (I.) and (II.) above are equivalent. The 'equation x − y = 1 of (I.) is replaced by the equivalent equation 2x-2y= 2. (ii.) If any equation of a system be replaced by an equation obtained by adding or subtracting corresponding members of two or more of the equations of the system, the resulting system will be equivalent to the given one. Thus, the system (II.) is equivalent to the system (iii.) If one equation of a system be solved for one of the unknown numbers, and the resulting value be substituted for this unknown number in each of the other equations, the derived system will be equivalent to the given one. E.g., the systems x - y = 2, 5x-3y=12, are equivalent. } (IV.) and x = 2 + y, } (v.) 5(2+y)-3y= 12, The proofs of the principles enunciated are as follows: (i.) Let A B, and C = D, = (I.) be two equations in two unknown numbers, say x and y; and let C' = D' be equivalent to C = D. Then the system = A B, and C' = D', (II.) is equivalent to the system (I.). For, by definition of equivalent equations, the same sets of values which satisfy C = D also satisfy C' = D', and vice versa. Therefore, any one of these sets of values, which also satisfies AB, is a solution of both systems. Consequently, every solution of either system is a solution of the other. In like manner, the principle can be proved for a system of any num ber of equations. (ii.) The proof is very similar to that of Ch. IV., § 3, Art. 5 (i.). (iii.) Let = A B, (1) and C = D, (2) (I.) P be two equations in two unknown numbers, say x and y; and let x = be the equation derived by solving (1) for x, and C'D' be the equation obtained by substituting P for x in (2). Then the system x = P, (3) and C' = D', (4) is equivalent to the system (I.). (II.) Since equation (3) is equivalent to equation (1), any solution of the system (1) must satisfy equation (3); that is, must give to x and P one and the same value. But (4) differs from (2) only in having P where (2) has x. Therefore, since x and P have the same value, any value of x, with the corresponding value of y, which makes C and D equal must make C' and D' equal. Therefore, every solution of the system (I.) is a solution of the system (II.). Since equation (1) is equivalent to equation (3), any solution of the system (II.) must satisfy equation (1); that is, must make A and B equal. But (2) differs from (4) only in having x where (4) has P. Therefore, since any solution of (II.) makes x and P equal and C' and D' equal, it must also make C and D equal. Therefore, every solution of the system (II.) is a solution of the system (I.). Consequently, the two systems are equivalent. In like manner, the principle can be proved for a system of any number of equations. 3. Elimination is the process of deriving from two or more equations of a system an equation with one less unknown number than the equations from which it is derived. The unknown number which does not appear in the derived equation is said to have been eliminated. in which the unknown number y does not appear. We say that y has been eliminated from the given equations. § 3. SYSTEMS OF LINEAR EQUATIONS. Linear Equations in Two Unknown Numbers. 1. There are several methods for solving two simultaneous equations in two unknown numbers. The object in all of them is to obtain from the given system an equivalent system of which one equation contains only one unknown number. Elimination by Addition and Subtraction. 2. Ex. 1. Solve the system 3x + 4y = 24, (1) 5x-6y=2. (2) (I.) To eliminate x, we multiply both members of equation (1) by 5, and both members of equation (2) by 3, thereby making the coefficients of x in the two equations equal. We then have The system (II.) is equivalent to the system (I.), by § 2, Art. 2 (i.). The system (II.) is, by § 2, Art. 2 (i.) and (ii.), equivalent to the system 3x+4y = 24, (15x+20 y) – (15 x 18 y) = 120 - 6; (5) or, performing the indicated operations, to (1) (III.) The system (V.) gives the required solution, since equation (7) gives the value of y, and equation (1) the corresponding value |