CHAPTER VII. SIMPLE EQUATIONS. 72. An equation asserts that two expressions are equal, but we do not usually employ the word equation in so wide a sense. Thus the statement x+3+x=2x+3, which is always true whatever value x may have, is called an identical equation, or an identity. The sign of identity frequently used is E. The parts of an equation to the right and left of the sign of equality are called members or sides of the equation, and are distinguished as the right side and left side. x= 73. Certain equations are only true for particular values of the symbols employed. Thus 3 x = 6 is only true when 2, and is called an equation of condition, or more usually an equation. Consequently an identity is an equation which is always true whatever be the values of the symbols involved; whereas an equation, in the ordinary use of the word, is only true for particular values of the symbols. In the above example 3 x = 6, the value 2 is said to satisfy the equation. The object of the present chapter is to explain how to treat an equation of the simplest kind in order to discover the value which satisfies it. . 74. The letter whose value it is required to find is called the unknown quantity. The process of finding its value is called solving the equation. The value so found is called the root or the solution of the equation, or "zero" of the equa 75. The solution of equations, and the operations sub sidiary to it, form an extremely important part of Mathe matics. All sorts of mathematical problems consist in the indirect determination of some quantity by means of its relations to other quantities which are known, and these relations are all expressed by means of equations. The operation in general of solving a problem in Mathematics, other than a transformation, is first, to express the conditions of the problem by means of one or more equations, and secondly, to solve these equations. For example, the problem which is expressed by the equation above given is the very simple question, "What is the number such that if multiplied by 3, the product is 6?" In the present chapter, it is the second of these two operations, the solution of an equation, that is considered. 76. An equation which involves the unknown quantity in the first degree is called a simple equation. The process of solving a simple equation depends upon the following axioms: 1. If to equals we add equals, the sums are equal. 2. If from equals we take equals, the remainders are equal. 3. If equals are multiplied by equals, the products are equal. 4. If equals are divided by equals, the quotients are equal. 77. Consider the equation 7 x = 14. It is required to find what numerical value x must have consistent with this statement. Dividing both sides by 7, we get Again, in the equation 7x-2x-x23+15-10, by TRANSPOSITION OF TERMS. 78. To solve 3x−8=x+ 12. Here the unknown quantity occurs on both sides of the equation. We can, however, transpose any term from one side to the other by simply changing its sign. This we proceed to show. Subtract x from both sides of the equation, and we get Thus we see that and appears as 3 x x = 12 +8 (Axiom 2). +x has been removed from one side, on the other; and - 8 has been removed from one side and appears as + 8 on the other. It is evident that similar steps may be employed in all Hence we may enunciate the following rule: cases. Rule. Any term may be transposed from one side of the equation to the other by changing its sign. 79. We may change the sign of every term in an equation; for this is equivalent to multiplying both sides by -1, which does not destroy the equality (Axiom 3). Ex. Take the equation - 3 x 12 x 24. which is the original equation with the sign of every term changed. 80. We can now give a general rule for solving a simple equation with one unknown quantity. Rule. Transpose all the terms containing the unknown quantity to one side of the equation, and the known quantities to the other. Collect the terms on each side; divide both sides by the coefficient of the unknown quantity, and the value required is obtained. Ex. 1. Solve 5(x − 3) − 7 (6 − x) + 3 = 24 − 3(8 − x). Removing brackets, 5 x — 15 – 42 + 7 x + 3 = 24 − 24 + 3 x ; Ex. 2. Solve 5x − (4 x − 7) (3 x − 5) = 6 – 8(4 x − 9)(x − 1,. 5 x − (12 x2 – 41 x + 35) = 6 − 3(4 x2 – 13 x + 9), Erase the term - 12x2 on each side and transpose; thus collecting terms, 5x+41x-39 x = 6 − 27 + 85; 7x=14. .. x=2. NOTE. Since the sign before a bracket affects every term within it, in the first line of work of Ex. 2, we do not remove the brackets until we have formed the products. Ex. 3. Solve 7 x 5[x-{7-6(x − 3)}]=3x+1. Removing brackets, we have transposing, collecting terms, 7 x − 5[x — {7 — 6x + 18}]= 3x + 1, ... x = 4. 81. It is extremely useful for the beginner to acquire the habit of occasionally verifying, that is, proving the truth of his results. Proofs of this kind are interesting and convincing; and the habit of applying such tests tends to make the student self-reliant and confident in his own accuracy. In the case of simple equations we have only to show that when we substitute the value of x in the two sides of the equation we obtain the same result. Ex. To show that x = 2 satisfies the equation 5x (4x-7)(3 x − 5) = 6 − 3(4 x − 9) (x − 1). When x=2, the left side 5x-(4x-7)(3x-5)=10-(8-7)(6-5) The right side 6 – 3(4 x − 9) (x − 1) = 6 – 3(8 − 9) (2 − 1) Thus, since these two results are the same, x = 2 satisfies the ¿quation. 8. 8(x-3)-(6-2x)=2(x+2) — 5(5 −x). 9. 7(25 - x) — 2 x = 2(3 x − 25). 18. 5x (3x-7)-(4-2x-(6x-3)} = 10. 19. 14x(5x − 9) - {4 – 3 x − (2 x − 3)} = 30. - 20. 25x19 [3 - {4x-5}]=3x-(6x-5). 21. (x+1)(2x+1)=(x+3) (2x+3) — 14. 1 22. (x+1)2(x2 − 1) = x(2x + 1) −2(x+2) (x + 1) + 20. 23. 2(x+1)(x + 3) + 8 = (2 x + 1)(x + 5). 24. 24. 6(x2-3x+2)-2(x2-1)=4(x+1)(x+2)- 28. 20(2x)+3(x − 7) − 2[x + 9-3 (9-4(2-x)}]= 22. 29. x + 2 −[ – 8 – 2 {8 – 3(5 – x)_ }]=0. 30. 3(5-6x) — 5[x − 5{1 – 3(x − 5)}]= 23. 31. (x+1)(2x+3)=2(x + 1)2 + 8. - 32. 3(x-1)2− 3 (x2 − 1) = x — 15. 33. (3x+1)(2x-7)=6(x-3)2+7. 34. x2-8x+25= x(x − 4) — 25 (x — 5) — 16. 35. x(x+1)+(x + 1)(x+2) = (x + 2)(x+3)+x(x+4) − & 36. 2(x+2)(x-4)= x(2x+1)- 21. 37. (x+1)2+2(x+3)2 = 3x(x+2)+35. 38. 4(x+5)2- (2 x + 1)2 = 3(x-5)+ 180. 39. 84+(x+4)(x − 3)(x+5)=(x + 1)(x+2)(x+3). 40. (x+1)(x+2)(x+6)=x8 +9x2+4(7x-1). |