Ans. ax-a2-bx+ab—ax+a2-bx+ab=2nb. SOLUTION OF EQUATIONS OF THE FIRST DEGREE, CONTAINING ONLY ONE UNKNOWN QUANTITY. ART. 152. The unknown quantity in an equation may be combined with the known quantities, either by addition, subtraction, multiplication, or division; or by two or more of these different methods. 1. Let it be required to find the value of x, in the equation a+x=b, where the unknown quantity is connected by addition. By subtracting a from each side (Art. 148), we have x-b-a. 2. Let it be required to find the value of x, in the equation x-a-b, where the unknown quantity is connected by subtraction. By adding a to each side (Art. 148), we have x=b+a. 3. Let it be required to find the value of x, in the equation ax=b, where the unknown quantity is connected by multiplication. By dividing each side by a, we have b x= 4. Let it be required to find the value of x, in the equation x=b, a where the unknown quantity is connected by division. By multiplying each side by a, we have x=bxa=ab From the solution of these examples, we see that When the unknown quantity is connected by addition, it is to be separated by subtraction. When it is connected by subtraction, it is to be separated by addition. When it is connected by multiplication, it is to be separated by division. And, when it is connected by division, it is to be separated by multiplication. 5. Let it be required to find the value of x, in the equation Clearing the equation of fractions, we have 21x-(24-2x)=7x+56, or 21x-24+2x=7x+56. Transposing the terms 7x and -24, we have 21x+2x-7x=56-+-24; reducing, 16x=80; dividing by 16, x=18-5. It will be readily seen that this solution consists of three steps, viz.: 1st. Clearing the equation of fractions. 2nd. Transposition. 3rd. Reducing like terms, and dividing by the coëfficient of x. Let this value of x be substituted instead of x in the original equation, and, if it is the true value, the two members will be equal to each other. The operation of substituting the value of the unknown quantity instead of itself, in the original equation, to see if it will render the two members equal to each other, is called verification. 6. Find the value of x, in the equation ART. 153. From the solution of the preceding examples, we derive the following RULE FOR THE SOLUTION OF AN EQUATION OF THE FIRST DEGREE.1. If necessary, clear the equation of fractions; and perform all the operations indicated. 2. Transpose all the terms containing the unknown quantity to one side, and the known quantities to the other. 3. Reduce each member to its simplest form, and divide both sides by the coefficient of the unknown quantity. REMARK. This rule gives the method of proceeding most generally advantageous, but in some cases it is best to perform the operations indicated, and transpose the necessary terms, before clearing of fractions. Experience can alone determine the best method in particular cases. EXAMPLES FOR PRACTICE. NOTE.-Let the pupil verify the value of the unknown quantity in each example. Find the value of the unknown quantity in each of the following examples. 3x+7 2x-7 18x+42-8x+28+231—21x-84; 7. VERIFICATION. -11x=-385, 8. 5(x+1)-2-3(x+5). 10. 5-3(4-x)+4(3-2x)=0. Ans. x=1. 11. 3(x-3)-2(x-2)+x-1=x+3+2(x+2)+3(x+1). Ans. x=-4. 12. 5(5x-6)—4(4x-5)+3(3x-2)-2x-16-0. 21. (2x-10)—††(3x-40)=15-1(57-x). Ans. x=17. 37. }(x—a)—}(2x—3b)—1 (a—x)=10a+11b. 3x-a, x+2b_7x 262 38. + Ans. x= b 12(26-c.) QUESTIONS PRODUCING EQUATIONS OF THE FIRST DEGREE, CONTAINING ONLY ONE UNKNOWN QUANTITY. ART. 154. The solution of a problem by algebra, consists of two distinct parts: 1st. To express the conditions of the problem in algebraic language; that is, to form the equation. 2nd. To solve the equation; that is, to find the value of the unknown quantity. Sometimes the statement of the question proposed, furnishes the equation directly; and sometimes it is necessary, from the conditions given, to deduce others, from which to form the equation. When the conditions furnish the equation directly, they are called explicit conditions. When the conditions are deduced from those given in the question, they are called implied conditions. It is impossible to give a precise rule by means of which every question may be readily stated in the form of an equation. The first step is, to understand fully the nature of the question, so as to be able to prove the correctness or incorrectness of any proposed answer. After this, the equation, by the solution of which the value of the unknown quantity is to be found, may generally be formed by the following RULE. Denote the required quantity by one of the final letters of the alphabet; then, by means of signs, indicate the same operations that it would be necessary to make on the answer, to verify it. |