Explanation. In the first step we observe that the-2, being transposed, becomes +2; in the second step +5, being transposed, becomes-5, &c. it must be likewise observed, that transposing does not affect the equality; the quantities on one side the = being equal to those on the other, as well after transposing, as before: thus, in the given equation, the aggregate of the numbers on each side the = is 7; in the first step it is 9, in the second 4, in the third-2, in the fourth -5, in the fifth-12, in the sixth —3, in the seventh -5, and in the eighth O nothing; in every step the sum of the numbers on one side the =, incorporated together according to their signs, is equal to that of the numbers so incorporated on the other. 75. When several quantities are to be transposed, it is not necessary to take them one at a time, as in the foregoing operation; they may be transposed all together, observing to change the sign of each of them. 2. Given a+b-c=d, to transpose b—c a. Thus a=d-b+c, the answer. 3. Given x=b-z+y—4, to transpose ―z+y—4. Thus x+z−y+4=b, the answer. 4. Given a+b-c=3-y, to transpose +b-c and-y. Thus, to transpose+b-c....a=3—y—b+c. And to transpose-y....a+y=3—b+c. 76. The unknown quantity in an equation being connected with known ones by the sign + or - to find its value. RULE. Transpose all the known quantities which are connected with the unknown one, and (Art. 75.) collect them together into one, according to their signs; the result will be the value of the unknown quantity". d This operation (as was observed in the preceding note) is equivalent to adding-b+c to both sides: thus to the given equation a+b—c=d the sum is a = d-b+c, the ans. Again, in ex. 3. to the given equation x = 4—2+y-4 and the same of other examples. the sum is x+z−y+4= b • In the foregoing rules we are taught how transposition is performed, here we learn its object; namely, to get the unknown quantity by itself on one side of the equation. In the operation of ex. 5. we have r=4-1+2−3; now 4-12-3 equals 6-4, and 6-4 equals 2, therefore r equals 2, because "things that are equal to the same are evidently equal to one another." 5. Given x+1-2+3=4, to find the value of x. OPERATION. x=4-1+2-3=6-4=2. Answer. Explanation. Having transposed + 1-2+3, they become on the opposite side-1+2-3'; whence 4-1+2-3, where + 4 and + 2 added together give +6, and i and-3 give-4: whence 6-4, or 2, is the answer, or value of x. 6. Given x-—1—8+2=3, to find the value of x. 7. Given x+5=6, to find x. Ans. x=1. Ans. x=11. 8. Given x-5=6, to find x. 9. Given x+1-2=3, to find x. 10. Given x8+7-6=5, to find r. Ans. x=4. Ans. x=12. 77. When the unknown quantity is negative. RULE. Transpose the unknown quantity, and proceed as before; or if, after the known quantities are transposed, the unknown one be negative, change the signs of all the terms of the equation . f To prove that the number found by the operation is really the value of the unknown quantity, and that no other number can possibly be its value, this is the rule. Substitute the value found instead of the unknown quantity in the equation proposed, and if by the adding, subtracting, &c. of the other quantities connected, according to the import of their signs, the result come out the same as the given one, the number found is the value of the unknown quantity: thus, ex. 6. If the number found, viz. 10, be substituted instead of x in the given equation x-1-8+2=3, it will become 10-1-8+2=3. Also ex. 7. If 1 be substituted for x, the given equation becomes 1+5=6; and it may be remarked, that if any number greater or less than the proper answer be substituted, it will give a result different from that proposed in the equation, and consequently shew that the number substituted is not the true answer. See note at the bottom of the following page. It is usual to retain the unknown quantity on the left side of the equation, and to transpose all the known quantities to the right; and then, if the unknown quantity happen to be negative, the sign of every term of the equation is to be changed, in order to make it affirmative. This change of signs is equivalent to transposing all the terms of the equa tion, and therefore depends on the same reasons as Art. 74. Otherwise, 11. Given 6=14-x, to find the value of x. Thus, by transposing-x....6+x=14. And by transposing +6 x=14-6=8, the ans. Or, by changing the signs of all the terms of the given equation, we shall have—6——14+x; whence transposing—14, we have —6 +14=x, or 8=x, as before. 13. Given 2-x=3, to find x. Ans. x=-1. 14. Given 10-x=11-5, to find x. Ans. x=4. 78. When the unknown quantity has a coefficient or multiplier. RULE 1. Transpose all the known quantities to the opposite side, and collect them together as in the former rules. II. Take away the coefficient from the unknown quantity, and divide the sum of the known ones by it; the quotient will be the value of the unknown quantity ↳. 15. Given 3x-6=9, to find the value of x. By transposing-6 we have 3 x=(9+6=) 15. Now, by taking away the coefficient 3, and dividing 15 by it, we have x= 15 3 =) 5 i, the answer required. When the unknown quantity is found in two terms, and on opposite sides of the equation, if you would make the unknown quantity affirmative, observe, that, 1. When both terms containing the unknown quantity are affirmative, the least must be transposed. 2. When both terms are negative, the greatest must be transposed. 3. When one term is affirmative, and the other negative, the negative term must be transposed. h" If equals be divided by equals, the quotients will be equal;" this selfevident truth is the foundation of the rule; for taking away the coefficient from the unknown quantity, is the same as dividing the term which contains it, by that coefficient; wherefore, taking away the said coefficient, and dividing the rest of the equation by it, is dividing equals by the same; consequently the results will be equal. i Proof of ex. 15. It is affirmed, that x=5 for then 3 = 3 × 5=15 and 3x-6=15-6=9, as in the example. If any num-ber less than 5 be put for x, suppose 4, then the result will be 3 X 4-6-12 16. Given 4y+5-6-7, to find y. By transposing+5-6, we have 4 y= (7-5+6=) 8. 20. Given 5 y-6+4=11, to find y. Ans. y=2 79. When the unknown quantity has a divisor. RULE I. Transpose all the known quantities to the opposite side, and collect them together as before. II. Take away the divisor, and multiply the sum of the known quantities by it; the product will be the value of the unknown quantity'. 22. Given +4-5=7, to find the value of x. X 3 By transposing+4—5, we get —— = (7—4+5=) 8. 3 Then taking away the divisor 3, and multiplying 8 by it, we shall have x=(3x8=) 24, the answer TM. —6=6, too little. Let a greater number, as 6, be proposed; then 3 × 6—6— 18-6-12, too much for the result proposed is 9; wherefore neither 4 nor 6 can equal, since one gives a result too little, and the other too much. k Proof of ex. 16. If 2 be substituted for y, the given equation becomes (2 × 4+5-6, or 8+5-6, or) 13-6-7, as was proposed. 1 This rule depends on a well known self-evident principle, namely," If equals be multiplied by equals, the products will be equal." Now taking away the divisor from the unknown quantity, is the same as multiplying the term containing it, by that divisor; then, if the divisor be taken away, and all the rest of the equation be multiplied by it, equals will be multiplied by the same, and consequently the results will be equal. m 24 Proof of ex. 22. By substituting 24 for a in the given equation, it becomes +4-5=8+4-5=) 12-5=7. Ex. 23. By substituting -12 for y, the equation becomes 12 · + 2 =) −1 + 2 = 1, as required. y 23. Given+2=1, to find y. By transposing+2, we obtain 12 = (1-2)-1. Whence multiplying by 12 .. y=(12x-1=)-12, the ans. So. When the unknown quantity has both a multiplier, and a divisor; that is, a fractional coefficient. RULE. Transpose, &c. as before; then take away the divisor by the preceding rule, and the multiplier by Art. 78. the result will be the answer. Or, To take away the whole fractional coefficient together; invert it, then multiply the known quantity by it, (so inverted,) the result will be the answer as before ". 29. Given -9=12, to find the value of x. 3x By transposition (Art. 76.) 3 x 4 (12+9=) 21. By multiplication (Art. 78.) 3x=(4x21) 84. By division (Art. 79.) x=( 84 =) 28°, the answer. "The taking away of the multiplier depends on Art. 78. and of the divisor on Art. 79. but if the learner chuses to take both away together, he must consider both together as forming a fractional coefficient; then the division of the rest of the equation by this fraction is evidently performed by inverting it, and multiplying, agreeably to the rule for division of fractions. Art. 204. Part I. X • Proof ( 4 8 -9=3×7—9=) 21-9=12, as required. |