Or, briefly : 96. Both members of an equation may be multiplied, or divided, by the same number. By Axiom 5 If A = B and B = D; we have, A = D. THE TRANSPOSITION OF TERMS 97. Most equations are given in such a form that the known and the unknown terms occur together in both members. Transposition is the process of changing the form of an equation so that the unknown terms shall all be in one member, usually the left, and the known terms all in the other. The process is based on Art. 95. In (3) we find c in the right member with its sign changed from to +. In (3) we find be in the left member with its sign changed from + to In general: 98. Any term in an equation may be transposed from one member to the other member if its sign is changed. As a direct consequence of the use of the axioms, we have: (1) The same term with the same sign in both members of an equation may be discarded. Given the equation, 3x + a − n = 2 x + a + m. (2) The sign of every term in an equation may be changed without destroying the equality. The sign of a root in a solution depends upon the law of signs for division. Thus: 99. When both members of an equation are reduced to simplest form, like signs in both members give a positive root, unlike signs a negative root. If the coefficient of the unknown quantity in a simplified equation is not exactly contained in the known quantity, the root is a fraction; and if, in a simplified equation, the member containing the known quantities reduces to zero, the root is THE VERIFICATION OF LINEAR EQUATIONS 100. To substitute a root in an equation is to replace the unknown literal factor in each term by the value of the root obtained. 101. To verify a root is to show that, by the substitution of this value, the given equation reduces to an identity. The verification of a root, as illustrated in the solutions following, should always be made in the original equation. THE GENERAL SOLUTION OF THE LINEAR EQUATION Illustrations: 1. Solve 5x-4=3x+12. 5x-4 = 3x + 12. Transposing 3 x to the left member and — 4 to the right member, Uniting terms, 5x-3x=4 + 12. 2 x = 16. Therefore, 8 is the correct value of the root, for, by substituting 8 for x in the original equation, we obtain an identity. Substitute 2 for x, 5(-2)-[8-(-2-2(-2)-1)]=- 10. Simplifying, 5x-[3-(x-2x-1)] 10. = 3. Solve (x+3)(2 x — 5)=2(x − 2)2 - 2(x + 1). Substituting for x in the original equation, (H+3)(t} -5)=2(}} − 2)2 — 2(}} + 1). (H)(-1)=2(-)2 -2({}). -797-781. = From the foregoing, we may state the general method: 102. Perform all indicated multiplications and remove all parentheses. Transpose the terms containing the unknown quantity to one member, and all known terms to the other member of the equation. Collect the terms in each member. Divide both members by the coefficient of the unknown quantity. · Exercise 16 (See also page 390.) Find and verify the roots of the following: 17. 2x-1+(4x-2)=1. 19. 3x-(2x+1)-(4x-3)=0. 20. 4x (5x+1)=8−(6x+9). 18. 3x+7-2(x+1)=6. 21. (2x+11) + 13 = 5 − (x + 1). 24. (x+7)-(3 x + 1) = 5 x + 4. 31. (x+1)(x+3)= (x − 2) (x − 5). 32. (2x-3)(x-7)=(x-1)(x+4)+x2. 33. (x-3)2+2 (x − 4)2 — 3 (x − 5)(x+5)= 7. 34. 5(x-1)-3(x-2)2=(2x-1) (3+x)-6. 35. 4[3x-2(x2+1)]= 7 — 4 x (2 x — 16). 36. —[2(x − 3)(x − 5) − (x+7) (3 − x)] = − 3 (x2 +3). 37. (x+2)3-(x − 1)3 — (3 x + 1) (3 x − 4) = 0. 38. 2.7x-(11-1.3 x) - 6.7 x = .62+4x-11. x 39. 0.007 -2(.0035x+.07) = .017-(.14.85 x). THE SOLUTION OF PROBLEMS 103. A problem is a question to be solved. In general, a problem is a statement of conditions involving an unknown number or numbers. We seek the value of that unknown number, and by assuming a literal symbol for the unknown we are able to state the given conditions in terms of that unknown. |