56. Linear equations in two variables. A simple equation in one variable has one and only one solution, as we have already seen (p. 33). On the other hand, an equation of the first degree in two variables has many solutions. is satisfied by innumerable pairs of numbers which may be substituted for x and y. For, transposing the term in y, we get from which it appears that when y has any particular numerical value the equation becomes a linear equation in x alone, and hence has a solution. Thus, when y = 1, x2, and this pair of values is a solution of the equation. Similarly, x = 9, y = 4 also satisfy the equation. 57. Solution of a pair of equations. If in solving the equation just considered, the values of x and y that one may use are no longer unrestricted in range, but must also satisfy a second linear equation, we get usually only a single pair of solutions. Thus if we seek a solution, that is, a pair of values of x and y satisfying such that also 3x + 7y = 1, x+y=-1, 1 satisfy both equa we find that the pair of values x = — tions. Any other solution of the first equation, as, for instance, x=9,y=-4, does not obey the condition imposed by the second. Two equations which are not reducible to the same form are called independent. are not independent, since the first is readily reduced to the second by transposing and dividing by 2. They are, in fact, essentially the same equation. On the other hand, and x-4y=2 3x-4y=2 are not reducible to the same form and are independent. Since dependent equations are identical except for the arrangement of terms and some constant factor, all their solutions are common to each other. Independent equations in more than one variable which have a common solution are called simultaneous equations. Two pairs of simultaneous equations which are satisfied by the same pair (or pairs) of values of x and y and only these are called equivalent. – 2, and x + y = − 1 = 1 are equivalent pairs of equations. 58. Independent equations. We now prove the following THEOREM. If A=0 and B=0 represent two independent equations, then the pairs of equations are equivalent where a, b, c, and d are any numbers such that The letters A and B symbolize linear expressions in x and y. Evidently any pair of values of x and y that makes both A = 0 and B = 0, i.e. satisfies (1), also makes aA +bB = 0 and cA + dB = 0, i.e. also satisfies (2). We must also show that any values of x and y that satisfy (2) also satisfy (1). For a certain pair of values of x and y let But ad be is not zero, by hypothesis; consequently B = 0. Similarly we could show that A = 0. B = Thus if we seek the solution of a pair of equations A = 0, 0, we may obtain by use of this theorem a pair of equivalent equations whose solution is evident, and find immediately the solution of the original equations. 59. Solution of a pair of simultaneous linear equations. The foregoing theorem affords the following RULE. Multiply each of the equations by some number such that the coefficients of one of the variables in the resulting pair of equations are identical. Subtract one equation from the other and solve the resulting simple equation in one variable. Find the value of the other variable by substituting the value just found in one of the original equations. Check the result by substituting the values found for both variables in the other equation. 60. Incompatible equations. Equations in more than one variable that do not have any common solution are called incompatible. THEOREM. The equations ax + by = C, a'x + b'y = c' are incompatible when and only when ab' - ba' = 0. (1) (2) Apply the rule of § 59 to find the solution of these equations. Multiply (1) by a' and (2) by a. a'b is not zero, we get a value of y; but since under our hypothesis ab' a'b = 0, we can get no value for y since division by zero is ruled out (§ 7). Thus no solution of (1) and (2) which is absurd. Thus no solution exists. 61. Résumé. We observe that pairs of equations of the form ax + by + c = 0, a'x + b'y + c' = 0 fall into three classes: (a) Dependent equations, which have innumerable common solutions. Then α b с (b) Incompatible equations, which have no common solution. Then ab' - a'b = 0, but (1) is not true. (1) (c) Simultaneous equations, which have one and only one pair of solutions. |